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ÀjÿÿÿHÄ [Ã ¸ ëðf.Êsò¸ ëâ HìHÄÃ at least at most exactly long an integer is required numpy/random/mtrand/mtrand.c %s (%s:%d) Missing type object numpy cannot import name %.230s name '%.200s' is not defined mtrand.pyx mtrand.RandomState.__reduce__ mtrand.RandomState.rand mtrand.RandomState.bytes randint_helpers.pxi mtrand._rand_uint64 mtrand._rand_int8 mtrand._rand_int32 mtrand._rand_int16 mtrand.RandomState.randn mtrand._shape_from_size mtrand.RandomState.__init__ mtrand.RandomState.get_state mtrand.disc0_array mtrand.RandomState.tomaxint mtrand.RandomState.set_state mtrand.discnmN_array_sc mtrand.cont2_array_sc mtrand.cont1_array_sc mtrand.cont0_array standard_cauchy standard_exponential standard_normal random_sample mtrand.cont3_array_sc float division mtrand.RandomState.dirichlet mtrand.discd_array_sc mtrand.RandomState.randint mtrand.discd_array mtrand.RandomState.logseries mtrand.RandomState.geometric mtrand.RandomState.zipf mtrand._rand_bool multinomial mtrand.cont1_array mtrand.RandomState.pareto mtrand.RandomState.chisquare mtrand.RandomState.rayleigh random_integers mtrand.RandomState.poisson mtrand.RandomState.seed hypergeometric mtrand.discnmN_array mtrand.RandomState.weibull mtrand._rand_int64 mtrand.discnp_array mtrand.discnp_array_sc mtrand.RandomState.binomial multivariate_normal mtrand.RandomState.shuffle standard_gamma mtrand.cont2_array mtrand.RandomState.wald mtrand.RandomState.logistic mtrand.RandomState.gumbel mtrand.RandomState.laplace mtrand.RandomState.vonmises noncentral_chisquare mtrand.RandomState.f mtrand.RandomState.gamma mtrand.RandomState.beta mtrand.RandomState.normal mtrand.RandomState.uniform mtrand.RandomState.lognormal mtrand.RandomState.power mtrand.RandomState.choice assignment mtrand.discdd_array mtrand.RandomState.standard_t negative_binomial mtrand.discdd_array_sc mtrand._rand_uint16 mtrand._rand_uint8 mtrand._rand_uint32 mtrand.cont3_array mtrand.RandomState.triangular noncentral_f %d.%d %s __builtin__ __builtins__ 2147483648 4294967295 4294967296 18446744073709551616 -9223372036854775808 __name__ numpy.pxd ndarray flatiter broadcast numpy.core.multiarray _ARRAY_API _ARRAY_API not found _ARRAY_API is NULL pointer init mtrand dtype mtrand.RandomState __getstate__ __setstate__ permutation %.200s() takes %.8s %zd positional argument%.1s (%zd given) need more than %zd value%.1s to unpack __%.4s__ returned non-%.4s (type %.200s) can't convert negative value to unsigned long value too large to convert to npy_int32 value too large to convert to npy_int16 value too large to convert to npy_int8 while calling a Python object NULL result without error in PyObject_Call can't convert negative value to npy_uint64 value too large to convert to npy_uint32 can't convert negative value to npy_uint32 value too large to convert to npy_uint16 can't convert negative value to npy_uint16 value too large to convert to npy_uint8 can't convert negative value to npy_uint8 value too large to convert to npy_bool can't convert negative value to npy_bool can't convert negative value to size_t value too large to convert to int Cannot convert %.200s to %.200s too many values to unpack (expected %zd) %.200s.%.200s is not a type object %s.%s size changed, may indicate binary incompatibility. Expected %zd, got %zd %.200s.%.200s has the wrong size, try recompiling. Expected %zd, got %zd %.200s() keywords must be strings %.200s() got an unexpected keyword argument '%.200s' %s() got multiple values for keyword argument '%s' raise: arg 3 must be a traceback or None instance exception may not have a separate value raise: exception class must be a subclass of BaseException mtrand.RandomState.__getstate__ mtrand.RandomState.__setstate__ '%.200s' object is unsliceable mtrand.RandomState.standard_cauchy mtrand.RandomState.standard_exponential mtrand.RandomState.standard_normal mtrand.RandomState.random_sample mtrand.RandomState.permutation mtrand.RandomState.multinomial mtrand.RandomState.exponential mtrand.RandomState.random_integers mtrand.RandomState.hypergeometric mtrand.RandomState.multivariate_normal mtrand.RandomState.standard_gamma mtrand.RandomState.noncentral_chisquare '%.200s' object does not support slice %.10s mtrand.RandomState.negative_binomial mtrand.RandomState.noncentral_f compiletime version %s of module '%.100s' does not match runtime version %s numpy.core.multiarray failed to import _ARRAY_API is not PyCObject object module compiled against ABI version 0x%x but this version of numpy is 0x%x module compiled against API version 0x%x but this version of numpy is 0x%x FATAL: module compiled as unknown endian FATAL: module compiled as little endian, but detected different endianness at runtime
RandomState(seed=None)
Container for the Mersenne Twister pseudo-random number generator.
`RandomState` exposes a number of methods for generating random numbers
drawn from a variety of probability distributions. In addition to the
distribution-specific arguments, each method takes a keyword argument
`size` that defaults to ``None``. If `size` is ``None``, then a single
value is generated and returned. If `size` is an integer, then a 1-D
array filled with generated values is returned. If `size` is a tuple,
then an array with that shape is filled and returned.
*Compatibility Guarantee*
A fixed seed and a fixed series of calls to 'RandomState' methods using
the same parameters will always produce the same results up to roundoff
error except when the values were incorrect. Incorrect values will be
fixed and the NumPy version in which the fix was made will be noted in
the relevant docstring. Extension of existing parameter ranges and the
addition of new parameters is allowed as long the previous behavior
remains unchanged.
Parameters
----------
seed : {None, int, array_like}, optional
Random seed used to initialize the pseudo-random number generator. Can
be any integer between 0 and 2**32 - 1 inclusive, an array (or other
sequence) of such integers, or ``None`` (the default). If `seed` is
``None``, then `RandomState` will try to read data from
``/dev/urandom`` (or the Windows analogue) if available or seed from
the clock otherwise.
Notes
-----
The Python stdlib module "random" also contains a Mersenne Twister
pseudo-random number generator with a number of methods that are similar
to the ones available in `RandomState`. `RandomState`, besides being
NumPy-aware, has the advantage that it provides a much larger number
of probability distributions to choose from.
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Þþÿ|ÞþÿsÞþÿjÞþÿaÞþÿy ÿÿp ÿÿg ÿÿ^ ÿÿU ÿÿ This function is deprecated. Please call randint({low}, {high} + 1) instead /home/charris/Workspace/numpy.git/numpy/random/mtrand/mtrand.pyx size is not compatible with inputs probabilities are not non-negative mean and cov must have same length covariance is not positive-semidefinite. cov must be 2 dimensional and square check_valid must equal 'warn', 'raise', or 'ignore' a must be 1-dimensional or an integer This function is deprecated. Please call randint(1, {low} + 1) instead RandomState.triangular (line 3592) RandomState.standard_t (line 2445) RandomState.standard_normal (line 1514) RandomState.standard_exponential (line 1779) RandomState.standard_cauchy (line 2381) RandomState.random_sample (line 814) RandomState.random_integers (line 1417) RandomState.permutation (line 4847) RandomState.noncentral_f (line 2099) RandomState.noncentral_chisquare (line 2277) RandomState.negative_binomial (line 3802) RandomState.multinomial (line 4530) Fewer non-zero entries in p than size Cannot take a larger sample than population when 'replace=False'
zipf(a, size=None)
Draw samples from a Zipf distribution.
Samples are drawn from a Zipf distribution with specified parameter
`a` > 1.
The Zipf distribution (also known as the zeta distribution) is a
continuous probability distribution that satisfies Zipf's law: the
frequency of an item is inversely proportional to its rank in a
frequency table.
Parameters
----------
a : float or array_like of floats
Distribution parameter. Should be greater than 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Zipf distribution.
See Also
--------
scipy.stats.zipf : probability density function, distribution, or
cumulative density function, etc.
Notes
-----
The probability density for the Zipf distribution is
.. math:: p(x) = \frac{x^{-a}}{\zeta(a)},
where :math:`\zeta` is the Riemann Zeta function.
It is named for the American linguist George Kingsley Zipf, who noted
that the frequency of any word in a sample of a language is inversely
proportional to its rank in the frequency table.
References
----------
.. [1] Zipf, G. K., "Selected Studies of the Principle of Relative
Frequency in Language," Cambridge, MA: Harvard Univ. Press,
1932.
Examples
--------
Draw samples from the distribution:
>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> from scipy import special
Truncate s values at 50 so plot is interesting:
>>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)
>>> plt.plot(x, y/max(y), linewidth=2, color='r')
>>> plt.show()
weibull(a, size=None)
Draw samples from a Weibull distribution.
Draw samples from a 1-parameter Weibull distribution with the given
shape parameter `a`.
.. math:: X = (-ln(U))^{1/a}
Here, U is drawn from the uniform distribution over (0,1].
The more common 2-parameter Weibull, including a scale parameter
:math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.
Parameters
----------
a : float or array_like of floats
Shape of the distribution. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Weibull distribution.
See Also
--------
scipy.stats.weibull_max
scipy.stats.weibull_min
scipy.stats.genextreme
gumbel
Notes
-----
The Weibull (or Type III asymptotic extreme value distribution
for smallest values, SEV Type III, or Rosin-Rammler
distribution) is one of a class of Generalized Extreme Value
(GEV) distributions used in modeling extreme value problems.
This class includes the Gumbel and Frechet distributions.
The probability density for the Weibull distribution is
.. math:: p(x) = \frac{a}
{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
where :math:`a` is the shape and :math:`\lambda` the scale.
The function has its peak (the mode) at
:math:`\lambda(\frac{a-1}{a})^{1/a}`.
When ``a = 1``, the Weibull distribution reduces to the exponential
distribution.
References
----------
.. [1] Waloddi Weibull, Royal Technical University, Stockholm,
1939 "A Statistical Theory Of The Strength Of Materials",
Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,
Generalstabens Litografiska Anstalts Forlag, Stockholm.
.. [2] Waloddi Weibull, "A Statistical Distribution Function of
Wide Applicability", Journal Of Applied Mechanics ASME Paper
1951.
.. [3] Wikipedia, "Weibull distribution",
http://en.wikipedia.org/wiki/Weibull_distribution
Examples
--------
Draw samples from the distribution:
>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()
vonmises(mu, kappa, size=None)
Draw samples from a von Mises distribution.
Samples are drawn from a von Mises distribution with specified mode
(mu) and dispersion (kappa), on the interval [-pi, pi].
The von Mises distribution (also known as the circular normal
distribution) is a continuous probability distribution on the unit
circle. It may be thought of as the circular analogue of the normal
distribution.
Parameters
----------
mu : float or array_like of floats
Mode ("center") of the distribution.
kappa : float or array_like of floats
Dispersion of the distribution, has to be >=0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``mu`` and ``kappa`` are both scalars.
Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized von Mises distribution.
See Also
--------
scipy.stats.vonmises : probability density function, distribution, or
cumulative density function, etc.
Notes
-----
The probability density for the von Mises distribution is
.. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},
where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
and :math:`I_0(\kappa)` is the modified Bessel function of order 0.
The von Mises is named for Richard Edler von Mises, who was born in
Austria-Hungary, in what is now the Ukraine. He fled to the United
States in 1939 and became a professor at Harvard. He worked in
probability theory, aerodynamics, fluid mechanics, and philosophy of
science.
References
----------
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing," New York: Dover, 1972.
.. [2] von Mises, R., "Mathematical Theory of Probability
and Statistics", New York: Academic Press, 1964.
Examples
--------
Draw samples from the distribution:
>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0
>>> plt.hist(s, 50, normed=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))
>>> plt.plot(x, y, linewidth=2, color='r')
>>> plt.show()
uniform(low=0.0, high=1.0, size=None)
Draw samples from a uniform distribution.
Samples are uniformly distributed over the half-open interval
``[low, high)`` (includes low, but excludes high). In other words,
any value within the given interval is equally likely to be drawn
by `uniform`.
Parameters
----------
low : float or array_like of floats, optional
Lower boundary of the output interval. All values generated will be
greater than or equal to low. The default value is 0.
high : float or array_like of floats
Upper boundary of the output interval. All values generated will be
less than high. The default value is 1.0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``low`` and ``high`` are both scalars.
Otherwise, ``np.broadcast(low, high).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized uniform distribution.
See Also
--------
randint : Discrete uniform distribution, yielding integers.
random_integers : Discrete uniform distribution over the closed
interval ``[low, high]``.
random_sample : Floats uniformly distributed over ``[0, 1)``.
random : Alias for `random_sample`.
rand : Convenience function that accepts dimensions as input, e.g.,
``rand(2,2)`` would generate a 2-by-2 array of floats,
uniformly distributed over ``[0, 1)``.
Notes
-----
The probability density function of the uniform distribution is
.. math:: p(x) = \frac{1}{b - a}
anywhere within the interval ``[a, b)``, and zero elsewhere.
When ``high`` == ``low``, values of ``low`` will be returned.
If ``high`` < ``low``, the results are officially undefined
and may eventually raise an error, i.e. do not rely on this
function to behave when passed arguments satisfying that
inequality condition.
Examples
--------
Draw samples from the distribution:
>>> s = np.random.uniform(-1,0,1000)
All values are within the given interval:
>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True
Display the histogram of the samples, along with the
probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, normed=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()
triangular(left, mode, right, size=None)
Draw samples from the triangular distribution over the
interval ``[left, right]``.
The triangular distribution is a continuous probability
distribution with lower limit left, peak at mode, and upper
limit right. Unlike the other distributions, these parameters
directly define the shape of the pdf.
Parameters
----------
left : float or array_like of floats
Lower limit.
mode : float or array_like of floats
The value where the peak of the distribution occurs.
The value should fulfill the condition ``left <= mode <= right``.
right : float or array_like of floats
Upper limit, should be larger than `left`.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``left``, ``mode``, and ``right``
are all scalars. Otherwise, ``np.broadcast(left, mode, right).size``
samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized triangular distribution.
Notes
-----
The probability density function for the triangular distribution is
.. math:: P(x;l, m, r) = \begin{cases}
\frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\
\frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\
0& \text{otherwise}.
\end{cases}
The triangular distribution is often used in ill-defined
problems where the underlying distribution is not known, but
some knowledge of the limits and mode exists. Often it is used
in simulations.
References
----------
.. [1] Wikipedia, "Triangular distribution"
http://en.wikipedia.org/wiki/Triangular_distribution
Examples
--------
Draw values from the distribution and plot the histogram:
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
... normed=True)
>>> plt.show()
tomaxint(size=None)
Random integers between 0 and ``sys.maxint``, inclusive.
Return a sample of uniformly distributed random integers in the interval
[0, ``sys.maxint``].
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : ndarray
Drawn samples, with shape `size`.
See Also
--------
randint : Uniform sampling over a given half-open interval of integers.
random_integers : Uniform sampling over a given closed interval of
integers.
Examples
--------
>>> RS = np.random.mtrand.RandomState() # need a RandomState object
>>> RS.tomaxint((2,2,2))
array([[[1170048599, 1600360186],
[ 739731006, 1947757578]],
[[1871712945, 752307660],
[1601631370, 1479324245]]])
>>> import sys
>>> sys.maxint
2147483647
>>> RS.tomaxint((2,2,2)) < sys.maxint
array([[[ True, True],
[ True, True]],
[[ True, True],
[ True, True]]], dtype=bool)
standard_t(df, size=None)
Draw samples from a standard Student's t distribution with `df` degrees
of freedom.
A special case of the hyperbolic distribution. As `df` gets
large, the result resembles that of the standard normal
distribution (`standard_normal`).
Parameters
----------
df : int or array_like of ints
Degrees of freedom, should be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``df`` is a scalar. Otherwise,
``np.array(df).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized standard Student's t distribution.
Notes
-----
The probability density function for the t distribution is
.. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}
The t test is based on an assumption that the data come from a
Normal distribution. The t test provides a way to test whether
the sample mean (that is the mean calculated from the data) is
a good estimate of the true mean.
The derivation of the t-distribution was first published in
1908 by William Gosset while working for the Guinness Brewery
in Dublin. Due to proprietary issues, he had to publish under
a pseudonym, and so he used the name Student.
References
----------
.. [1] Dalgaard, Peter, "Introductory Statistics With R",
Springer, 2002.
.. [2] Wikipedia, "Student's t-distribution"
http://en.wikipedia.org/wiki/Student's_t-distribution
Examples
--------
From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
women in Kj is:
>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
... 7515, 8230, 8770])
Does their energy intake deviate systematically from the recommended
value of 7725 kJ?
We have 10 degrees of freedom, so is the sample mean within 95% of the
recommended value?
>>> s = np.random.standard_t(10, size=100000)
>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727
Calculate the t statistic, setting the ddof parameter to the unbiased
value so the divisor in the standard deviation will be degrees of
freedom, N-1.
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(s, bins=100, normed=True)
For a one-sided t-test, how far out in the distribution does the t
statistic appear?
>>> np.sum(s<t) / float(len(s))
0.0090699999999999999 #random
So the p-value is about 0.009, which says the null hypothesis has a
probability of about 99% of being true.
rayleigh(scale=1.0, size=None)
Draw samples from a Rayleigh distribution.
The :math:`\chi` and Weibull distributions are generalizations of the
Rayleigh.
Parameters
----------
scale : float or array_like of floats, optional
Scale, also equals the mode. Should be >= 0. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``scale`` is a scalar. Otherwise,
``np.array(scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Rayleigh distribution.
Notes
-----
The probability density function for the Rayleigh distribution is
.. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}
The Rayleigh distribution would arise, for example, if the East
and North components of the wind velocity had identical zero-mean
Gaussian distributions. Then the wind speed would have a Rayleigh
distribution.
References
----------
.. [1] Brighton Webs Ltd., "Rayleigh Distribution,"
http://www.brighton-webs.co.uk/distributions/rayleigh.asp
.. [2] Wikipedia, "Rayleigh distribution"
http://en.wikipedia.org/wiki/Rayleigh_distribution
Examples
--------
Draw values from the distribution and plot the histogram
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, normed=True)
Wave heights tend to follow a Rayleigh distribution. If the mean wave
height is 1 meter, what fraction of waves are likely to be larger than 3
meters?
>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)
The percentage of waves larger than 3 meters is:
>>> 100.*sum(s>3)/1000000.
0.087300000000000003
random_integers(low, high=None, size=None)
Random integers of type np.int between `low` and `high`, inclusive.
Return random integers of type np.int from the "discrete uniform"
distribution in the closed interval [`low`, `high`]. If `high` is
None (the default), then results are from [1, `low`]. The np.int
type translates to the C long type used by Python 2 for "short"
integers and its precision is platform dependent.
This function has been deprecated. Use randint instead.
.. deprecated:: 1.11.0
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution (unless
``high=None``, in which case this parameter is the *highest* such
integer).
high : int, optional
If provided, the largest (signed) integer to be drawn from the
distribution (see above for behavior if ``high=None``).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : int or ndarray of ints
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
See Also
--------
random.randint : Similar to `random_integers`, only for the half-open
interval [`low`, `high`), and 0 is the lowest value if `high` is
omitted.
Notes
-----
To sample from N evenly spaced floating-point numbers between a and b,
use::
a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
Examples
--------
>>> np.random.random_integers(5)
4
>>> type(np.random.random_integers(5))
<type 'int'>
>>> np.random.random_integers(5, size=(3.,2.))
array([[5, 4],
[3, 3],
[4, 5]])
Choose five random numbers from the set of five evenly-spaced
numbers between 0 and 2.5, inclusive (*i.e.*, from the set
:math:`{0, 5/8, 10/8, 15/8, 20/8}`):
>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ])
Roll two six sided dice 1000 times and sum the results:
>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2
Display results as a histogram:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, normed=True)
>>> plt.show()
randint(low, high=None, size=None, dtype='l')
Return random integers from `low` (inclusive) to `high` (exclusive).
Return random integers from the "discrete uniform" distribution of
the specified dtype in the "half-open" interval [`low`, `high`). If
`high` is None (the default), then results are from [0, `low`).
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution (unless
``high=None``, in which case this parameter is one above the
*highest* such integer).
high : int, optional
If provided, one above the largest (signed) integer to be drawn
from the distribution (see above for behavior if ``high=None``).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
dtype : dtype, optional
Desired dtype of the result. All dtypes are determined by their
name, i.e., 'int64', 'int', etc, so byteorder is not available
and a specific precision may have different C types depending
on the platform. The default value is 'np.int'.
.. versionadded:: 1.11.0
Returns
-------
out : int or ndarray of ints
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
See Also
--------
random.random_integers : similar to `randint`, only for the closed
interval [`low`, `high`], and 1 is the lowest value if `high` is
omitted. In particular, this other one is the one to use to generate
uniformly distributed discrete non-integers.
Examples
--------
>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
Generate a 2 x 4 array of ints between 0 and 4, inclusive:
>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1],
[3, 2, 2, 0]])
power(a, size=None)
Draws samples in [0, 1] from a power distribution with positive
exponent a - 1.
Also known as the power function distribution.
Parameters
----------
a : float or array_like of floats
Parameter of the distribution. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized power distribution.
Raises
------
ValueError
If a < 1.
Notes
-----
The probability density function is
.. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.
The power function distribution is just the inverse of the Pareto
distribution. It may also be seen as a special case of the Beta
distribution.
It is used, for example, in modeling the over-reporting of insurance
claims.
References
----------
.. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions
in economics and actuarial sciences", Wiley, 2003.
.. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:
Dataplot Reference Manual, Volume 2: Let Subcommands and Library
Functions", National Institute of Standards and Technology
Handbook Series, June 2003.
http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf
Examples
--------
Draw samples from the distribution:
>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()
Compare the power function distribution to the inverse of the Pareto.
>>> from scipy import stats
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)
>>> plt.figure()
>>> plt.hist(rvs, bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('np.random.power(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of 1 + np.random.pareto(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of stats.pareto(5)')
pareto(a, size=None)
Draw samples from a Pareto II or Lomax distribution with
specified shape.
The Lomax or Pareto II distribution is a shifted Pareto
distribution. The classical Pareto distribution can be
obtained from the Lomax distribution by adding 1 and
multiplying by the scale parameter ``m`` (see Notes). The
smallest value of the Lomax distribution is zero while for the
classical Pareto distribution it is ``mu``, where the standard
Pareto distribution has location ``mu = 1``. Lomax can also
be considered as a simplified version of the Generalized
Pareto distribution (available in SciPy), with the scale set
to one and the location set to zero.
The Pareto distribution must be greater than zero, and is
unbounded above. It is also known as the "80-20 rule". In
this distribution, 80 percent of the weights are in the lowest
20 percent of the range, while the other 20 percent fill the
remaining 80 percent of the range.
Parameters
----------
a : float or array_like of floats
Shape of the distribution. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Pareto distribution.
See Also
--------
scipy.stats.lomax : probability density function, distribution or
cumulative density function, etc.
scipy.stats.genpareto : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Pareto distribution is
.. math:: p(x) = \frac{am^a}{x^{a+1}}
where :math:`a` is the shape and :math:`m` the scale.
The Pareto distribution, named after the Italian economist
Vilfredo Pareto, is a power law probability distribution
useful in many real world problems. Outside the field of
economics it is generally referred to as the Bradford
distribution. Pareto developed the distribution to describe
the distribution of wealth in an economy. It has also found
use in insurance, web page access statistics, oil field sizes,
and many other problems, including the download frequency for
projects in Sourceforge [1]_. It is one of the so-called
"fat-tailed" distributions.
References
----------
.. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
Sourceforge projects.
.. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
.. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
Values, Birkhauser Verlag, Basel, pp 23-30.
.. [4] Wikipedia, "Pareto distribution",
http://en.wikipedia.org/wiki/Pareto_distribution
Examples
--------
Draw samples from the distribution:
>>> a, m = 3., 2. # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m
Display the histogram of the samples, along with the probability
density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, normed=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()
normal(loc=0.0, scale=1.0, size=None)
Draw random samples from a normal (Gaussian) distribution.
The probability density function of the normal distribution, first
derived by De Moivre and 200 years later by both Gauss and Laplace
independently [2]_, is often called the bell curve because of
its characteristic shape (see the example below).
The normal distributions occurs often in nature. For example, it
describes the commonly occurring distribution of samples influenced
by a large number of tiny, random disturbances, each with its own
unique distribution [2]_.
Parameters
----------
loc : float or array_like of floats
Mean ("centre") of the distribution.
scale : float or array_like of floats
Standard deviation (spread or "width") of the distribution.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized normal distribution.
See Also
--------
scipy.stats.norm : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gaussian distribution is
.. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },
where :math:`\mu` is the mean and :math:`\sigma` the standard
deviation. The square of the standard deviation, :math:`\sigma^2`,
is called the variance.
The function has its peak at the mean, and its "spread" increases with
the standard deviation (the function reaches 0.607 times its maximum at
:math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that
`numpy.random.normal` is more likely to return samples lying close to
the mean, rather than those far away.
References
----------
.. [1] Wikipedia, "Normal distribution",
http://en.wikipedia.org/wiki/Normal_distribution
.. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
Random Variables and Random Signal Principles", 4th ed., 2001,
pp. 51, 51, 125.
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)
Verify the mean and the variance:
>>> abs(mu - np.mean(s)) < 0.01
True
>>> abs(sigma - np.std(s, ddof=1)) < 0.01
True
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
... np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
... linewidth=2, color='r')
>>> plt.show()
noncentral_f(dfnum, dfden, nonc, size=None)
Draw samples from the noncentral F distribution.
Samples are drawn from an F distribution with specified parameters,
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
freedom in denominator), where both parameters > 1.
`nonc` is the non-centrality parameter.
Parameters
----------
dfnum : int or array_like of ints
Parameter, should be > 1.
dfden : int or array_like of ints
Parameter, should be > 1.
nonc : float or array_like of floats
Parameter, should be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``dfnum``, ``dfden``, and ``nonc``
are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size``
samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized noncentral Fisher distribution.
Notes
-----
When calculating the power of an experiment (power = probability of
rejecting the null hypothesis when a specific alternative is true) the
non-central F statistic becomes important. When the null hypothesis is
true, the F statistic follows a central F distribution. When the null
hypothesis is not true, then it follows a non-central F statistic.
References
----------
.. [1] Weisstein, Eric W. "Noncentral F-Distribution."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/NoncentralF-Distribution.html
.. [2] Wikipedia, "Noncentral F-distribution",
http://en.wikipedia.org/wiki/Noncentral_F-distribution
Examples
--------
In a study, testing for a specific alternative to the null hypothesis
requires use of the Noncentral F distribution. We need to calculate the
area in the tail of the distribution that exceeds the value of the F
distribution for the null hypothesis. We'll plot the two probability
distributions for comparison.
>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, normed=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, normed=True)
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()
noncentral_chisquare(df, nonc, size=None)
Draw samples from a noncentral chi-square distribution.
The noncentral :math:`\chi^2` distribution is a generalisation of
the :math:`\chi^2` distribution.
Parameters
----------
df : int or array_like of ints
Degrees of freedom, should be > 0 as of NumPy 1.10.0,
should be > 1 for earlier versions.
nonc : float or array_like of floats
Non-centrality, should be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``df`` and ``nonc`` are both scalars.
Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized noncentral chi-square distribution.
Notes
-----
The probability density function for the noncentral Chi-square
distribution is
.. math:: P(x;df,nonc) = \sum^{\infty}_{i=0}
\frac{e^{-nonc/2}(nonc/2)^{i}}{i!}
\P_{Y_{df+2i}}(x),
where :math:`Y_{q}` is the Chi-square with q degrees of freedom.
In Delhi (2007), it is noted that the noncentral chi-square is
useful in bombing and coverage problems, the probability of
killing the point target given by the noncentral chi-squared
distribution.
References
----------
.. [1] Delhi, M.S. Holla, "On a noncentral chi-square distribution in
the analysis of weapon systems effectiveness", Metrika,
Volume 15, Number 1 / December, 1970.
.. [2] Wikipedia, "Noncentral chi-square distribution"
http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution
Examples
--------
Draw values from the distribution and plot the histogram
>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
... bins=200, normed=True)
>>> plt.show()
Draw values from a noncentral chisquare with very small noncentrality,
and compare to a chisquare.
>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
... bins=np.arange(0., 25, .1), normed=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
... bins=np.arange(0., 25, .1), normed=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()
Demonstrate how large values of non-centrality lead to a more symmetric
distribution.
>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
... bins=200, normed=True)
>>> plt.show()
negative_binomial(n, p, size=None)
Draw samples from a negative binomial distribution.
Samples are drawn from a negative binomial distribution with specified
parameters, `n` trials and `p` probability of success where `n` is an
integer > 0 and `p` is in the interval [0, 1].
Parameters
----------
n : int or array_like of ints
Parameter of the distribution, > 0. Floats are also accepted,
but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``n`` and ``p`` are both scalars.
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized negative binomial distribution,
where each sample is equal to N, the number of trials it took to
achieve n - 1 successes, N - (n - 1) failures, and a success on the,
(N + n)th trial.
Notes
-----
The probability density for the negative binomial distribution is
.. math:: P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N},
where :math:`n-1` is the number of successes, :math:`p` is the
probability of success, and :math:`N+n-1` is the number of trials.
The negative binomial distribution gives the probability of n-1
successes and N failures in N+n-1 trials, and success on the (N+n)th
trial.
If one throws a die repeatedly until the third time a "1" appears,
then the probability distribution of the number of non-"1"s that
appear before the third "1" is a negative binomial distribution.
References
----------
.. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/NegativeBinomialDistribution.html
.. [2] Wikipedia, "Negative binomial distribution",
http://en.wikipedia.org/wiki/Negative_binomial_distribution
Examples
--------
Draw samples from the distribution:
A real world example. A company drills wild-cat oil
exploration wells, each with an estimated probability of
success of 0.1. What is the probability of having one success
for each successive well, that is what is the probability of a
single success after drilling 5 wells, after 6 wells, etc.?
>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11):
... probability = sum(s<i) / 100000.
... print i, "wells drilled, probability of one success =", probability
multivariate_normal(mean, cov[, size, check_valid, tol])
Draw random samples from a multivariate normal distribution.
The multivariate normal, multinormal or Gaussian distribution is a
generalization of the one-dimensional normal distribution to higher
dimensions. Such a distribution is specified by its mean and
covariance matrix. These parameters are analogous to the mean
(average or "center") and variance (standard deviation, or "width,"
squared) of the one-dimensional normal distribution.
Parameters
----------
mean : 1-D array_like, of length N
Mean of the N-dimensional distribution.
cov : 2-D array_like, of shape (N, N)
Covariance matrix of the distribution. It must be symmetric and
positive-semidefinite for proper sampling.
size : int or tuple of ints, optional
Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because
each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
If no shape is specified, a single (`N`-D) sample is returned.
check_valid : { 'warn', 'raise', 'ignore' }, optional
Behavior when the covariance matrix is not positive semidefinite.
tol : float, optional
Tolerance when checking the singular values in covariance matrix.
Returns
-------
out : ndarray
The drawn samples, of shape *size*, if that was provided. If not,
the shape is ``(N,)``.
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
value drawn from the distribution.
Notes
-----
The mean is a coordinate in N-dimensional space, which represents the
location where samples are most likely to be generated. This is
analogous to the peak of the bell curve for the one-dimensional or
univariate normal distribution.
Covariance indicates the level to which two variables vary together.
From the multivariate normal distribution, we draw N-dimensional
samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix
element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
"spread").
Instead of specifying the full covariance matrix, popular
approximations include:
- Spherical covariance (`cov` is a multiple of the identity matrix)
- Diagonal covariance (`cov` has non-negative elements, and only on
the diagonal)
This geometrical property can be seen in two dimensions by plotting
generated data-points:
>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]] # diagonal covariance
Diagonal covariance means that points are oriented along x or y-axis:
>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()
Note that the covariance matrix must be positive semidefinite (a.k.a.
nonnegative-definite). Otherwise, the behavior of this method is
undefined and backwards compatibility is not guaranteed.
References
----------
.. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
Processes," 3rd ed., New York: McGraw-Hill, 1991.
.. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
Classification," 2nd ed., New York: Wiley, 2001.
Examples
--------
>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)
The following is probably true, given that 0.6 is roughly twice the
standard deviation:
>>> list((x[0,0,:] - mean) < 0.6)
[True, True]
multinomial(n, pvals, size=None)
Draw samples from a multinomial distribution.
The multinomial distribution is a multivariate generalisation of the
binomial distribution. Take an experiment with one of ``p``
possible outcomes. An example of such an experiment is throwing a dice,
where the outcome can be 1 through 6. Each sample drawn from the
distribution represents `n` such experiments. Its values,
``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the
outcome was ``i``.
Parameters
----------
n : int
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the ``p`` different outcomes. These
should sum to 1 (however, the last element is always assumed to
account for the remaining probability, as long as
``sum(pvals[:-1]) <= 1)``.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : ndarray
The drawn samples, of shape *size*, if that was provided. If not,
the shape is ``(N,)``.
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
value drawn from the distribution.
Examples
--------
Throw a dice 20 times:
>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]])
It landed 4 times on 1, once on 2, etc.
Now, throw the dice 20 times, and 20 times again:
>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3],
[2, 4, 3, 4, 0, 7]])
For the first run, we threw 3 times 1, 4 times 2, etc. For the second,
we threw 2 times 1, 4 times 2, etc.
A loaded die is more likely to land on number 6:
>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26])
The probability inputs should be normalized. As an implementation
detail, the value of the last entry is ignored and assumed to take
up any leftover probability mass, but this should not be relied on.
A biased coin which has twice as much weight on one side as on the
other should be sampled like so:
>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT
array([38, 62])
not like:
>>> np.random.multinomial(100, [1.0, 2.0]) # WRONG
array([100, 0])
logseries(p, size=None)
Draw samples from a logarithmic series distribution.
Samples are drawn from a log series distribution with specified
shape parameter, 0 < ``p`` < 1.
Parameters
----------
p : float or array_like of floats
Shape parameter for the distribution. Must be in the range (0, 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``p`` is a scalar. Otherwise,
``np.array(p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized logarithmic series distribution.
See Also
--------
scipy.stats.logser : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Log Series distribution is
.. math:: P(k) = \frac{-p^k}{k \ln(1-p)},
where p = probability.
The log series distribution is frequently used to represent species
richness and occurrence, first proposed by Fisher, Corbet, and
Williams in 1943 [2]. It may also be used to model the numbers of
occupants seen in cars [3].
References
----------
.. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional
species diversity through the log series distribution of
occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
Volume 5, Number 5, September 1999 , pp. 187-195(9).
.. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
relation between the number of species and the number of
individuals in a random sample of an animal population.
Journal of Animal Ecology, 12:42-58.
.. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
Data Sets, CRC Press, 1994.
.. [4] Wikipedia, "Logarithmic distribution",
http://en.wikipedia.org/wiki/Logarithmic_distribution
Examples
--------
Draw samples from the distribution:
>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> count, bins, ignored = plt.hist(s)
# plot against distribution
>>> def logseries(k, p):
... return -p**k/(k*log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
logseries(bins, a).max(), 'r')
>>> plt.show()
lognormal(mean=0.0, sigma=1.0, size=None)
Draw samples from a log-normal distribution.
Draw samples from a log-normal distribution with specified mean,
standard deviation, and array shape. Note that the mean and standard
deviation are not the values for the distribution itself, but of the
underlying normal distribution it is derived from.
Parameters
----------
mean : float or array_like of floats, optional
Mean value of the underlying normal distribution. Default is 0.
sigma : float or array_like of floats, optional
Standard deviation of the underlying normal distribution. Should
be greater than zero. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``mean`` and ``sigma`` are both scalars.
Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized log-normal distribution.
See Also
--------
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.
Notes
-----
A variable `x` has a log-normal distribution if `log(x)` is normally
distributed. The probability density function for the log-normal
distribution is:
.. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}
where :math:`\mu` is the mean and :math:`\sigma` is the standard
deviation of the normally distributed logarithm of the variable.
A log-normal distribution results if a random variable is the *product*
of a large number of independent, identically-distributed variables in
the same way that a normal distribution results if the variable is the
*sum* of a large number of independent, identically-distributed
variables.
References
----------
.. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal
Distributions across the Sciences: Keys and Clues,"
BioScience, Vol. 51, No. 5, May, 2001.
http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
.. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme
Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, normed=True, align='mid')
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
... / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()
Demonstrate that taking the products of random samples from a uniform
distribution can be fit well by a log-normal probability density
function.
>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
... a = 10. + np.random.random(100)
... b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, normed=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
... / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()
logistic(loc=0.0, scale=1.0, size=None)
Draw samples from a logistic distribution.
Samples are drawn from a logistic distribution with specified
parameters, loc (location or mean, also median), and scale (>0).
Parameters
----------
loc : float or array_like of floats, optional
Parameter of the distribution. Default is 0.
scale : float or array_like of floats, optional
Parameter of the distribution. Should be greater than zero.
Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized logistic distribution.
See Also
--------
scipy.stats.logistic : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Logistic distribution is
.. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},
where :math:`\mu` = location and :math:`s` = scale.
The Logistic distribution is used in Extreme Value problems where it
can act as a mixture of Gumbel distributions, in Epidemiology, and by
the World Chess Federation (FIDE) where it is used in the Elo ranking
system, assuming the performance of each player is a logistically
distributed random variable.
References
----------
.. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of
Extreme Values, from Insurance, Finance, Hydrology and Other
Fields," Birkhauser Verlag, Basel, pp 132-133.
.. [2] Weisstein, Eric W. "Logistic Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/LogisticDistribution.html
.. [3] Wikipedia, "Logistic-distribution",
http://en.wikipedia.org/wiki/Logistic_distribution
Examples
--------
Draw samples from the distribution:
>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> count, bins, ignored = plt.hist(s, bins=50)
# plot against distribution
>>> def logist(x, loc, scale):
... return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2)
>>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\
... logist(bins, loc, scale).max())
>>> plt.show()
hypergeometric(ngood, nbad, nsample, size=None)
Draw samples from a Hypergeometric distribution.
Samples are drawn from a hypergeometric distribution with specified
parameters, ngood (ways to make a good selection), nbad (ways to make
a bad selection), and nsample = number of items sampled, which is less
than or equal to the sum ngood + nbad.
Parameters
----------
ngood : int or array_like of ints
Number of ways to make a good selection. Must be nonnegative.
nbad : int or array_like of ints
Number of ways to make a bad selection. Must be nonnegative.
nsample : int or array_like of ints
Number of items sampled. Must be at least 1 and at most
``ngood + nbad``.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``ngood``, ``nbad``, and ``nsample``
are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size``
samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized hypergeometric distribution.
See Also
--------
scipy.stats.hypergeom : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Hypergeometric distribution is
.. math:: P(x) = \frac{\binom{m}{n}\binom{N-m}{n-x}}{\binom{N}{n}},
where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n`
for P(x) the probability of x successes, n = ngood, m = nbad, and
N = number of samples.
Consider an urn with black and white marbles in it, ngood of them
black and nbad are white. If you draw nsample balls without
replacement, then the hypergeometric distribution describes the
distribution of black balls in the drawn sample.
Note that this distribution is very similar to the binomial
distribution, except that in this case, samples are drawn without
replacement, whereas in the Binomial case samples are drawn with
replacement (or the sample space is infinite). As the sample space
becomes large, this distribution approaches the binomial.
References
----------
.. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
and Quigley, 1972.
.. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/HypergeometricDistribution.html
.. [3] Wikipedia, "Hypergeometric distribution",
http://en.wikipedia.org/wiki/Hypergeometric_distribution
Examples
--------
Draw samples from the distribution:
>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> hist(s)
# note that it is very unlikely to grab both bad items
Suppose you have an urn with 15 white and 15 black marbles.
If you pull 15 marbles at random, how likely is it that
12 or more of them are one color?
>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
# answer = 0.003 ... pretty unlikely!
/home/charris/Workspace/numpy.git/numpy/random/mtrand/randint_helpers.pxi
gumbel(loc=0.0, scale=1.0, size=None)
Draw samples from a Gumbel distribution.
Draw samples from a Gumbel distribution with specified location and
scale. For more information on the Gumbel distribution, see
Notes and References below.
Parameters
----------
loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Gumbel distribution.
See Also
--------
scipy.stats.gumbel_l
scipy.stats.gumbel_r
scipy.stats.genextreme
weibull
Notes
-----
The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
Value Type I) distribution is one of a class of Generalized Extreme
Value (GEV) distributions used in modeling extreme value problems.
The Gumbel is a special case of the Extreme Value Type I distribution
for maximums from distributions with "exponential-like" tails.
The probability density for the Gumbel distribution is
.. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
\beta}},
where :math:`\mu` is the mode, a location parameter, and
:math:`\beta` is the scale parameter.
The Gumbel (named for German mathematician Emil Julius Gumbel) was used
very early in the hydrology literature, for modeling the occurrence of
flood events. It is also used for modeling maximum wind speed and
rainfall rates. It is a "fat-tailed" distribution - the probability of
an event in the tail of the distribution is larger than if one used a
Gaussian, hence the surprisingly frequent occurrence of 100-year
floods. Floods were initially modeled as a Gaussian process, which
underestimated the frequency of extreme events.
It is one of a class of extreme value distributions, the Generalized
Extreme Value (GEV) distributions, which also includes the Weibull and
Frechet.
The function has a mean of :math:`\mu + 0.57721\beta` and a variance
of :math:`\frac{\pi^2}{6}\beta^2`.
References
----------
.. [1] Gumbel, E. J., "Statistics of Extremes,"
New York: Columbia University Press, 1958.
.. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
Values from Insurance, Finance, Hydrology and Other Fields,"
Basel: Birkhauser Verlag, 2001.
Examples
--------
Draw samples from the distribution:
>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
... * np.exp( -np.exp( -(bins - mu) /beta) ),
... linewidth=2, color='r')
>>> plt.show()
Show how an extreme value distribution can arise from a Gaussian process
and compare to a Gaussian:
>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
... a = np.random.normal(mu, beta, 1000)
... means.append(a.mean())
... maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, normed=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
... * np.exp(-np.exp(-(bins - mu)/beta)),
... linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
... * np.exp(-(bins - mu)**2 / (2 * beta**2)),
... linewidth=2, color='g')
>>> plt.show()
geometric(p, size=None)
Draw samples from the geometric distribution.
Bernoulli trials are experiments with one of two outcomes:
success or failure (an example of such an experiment is flipping
a coin). The geometric distribution models the number of trials
that must be run in order to achieve success. It is therefore
supported on the positive integers, ``k = 1, 2, ...``.
The probability mass function of the geometric distribution is
.. math:: f(k) = (1 - p)^{k - 1} p
where `p` is the probability of success of an individual trial.
Parameters
----------
p : float or array_like of floats
The probability of success of an individual trial.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``p`` is a scalar. Otherwise,
``np.array(p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized geometric distribution.
Examples
--------
Draw ten thousand values from the geometric distribution,
with the probability of an individual success equal to 0.35:
>>> z = np.random.geometric(p=0.35, size=10000)
How many trials succeeded after a single run?
>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
gamma(shape, scale=1.0, size=None)
Draw samples from a Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters,
`shape` (sometimes designated "k") and `scale` (sometimes designated
"theta"), where both parameters are > 0.
Parameters
----------
shape : float or array_like of floats
The shape of the gamma distribution. Should be greater than zero.
scale : float or array_like of floats, optional
The scale of the gamma distribution. Should be greater than zero.
Default is equal to 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``shape`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized gamma distribution.
See Also
--------
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gamma distribution is
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
where :math:`k` is the shape and :math:`\theta` the scale,
and :math:`\Gamma` is the Gamma function.
The Gamma distribution is often used to model the times to failure of
electronic components, and arises naturally in processes for which the
waiting times between Poisson distributed events are relevant.
References
----------
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/GammaDistribution.html
.. [2] Wikipedia, "Gamma distribution",
http://en.wikipedia.org/wiki/Gamma_distribution
Examples
--------
Draw samples from the distribution:
>>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps
>>> count, bins, ignored = plt.hist(s, 50, normed=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /
... (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()
f(dfnum, dfden, size=None)
Draw samples from an F distribution.
Samples are drawn from an F distribution with specified parameters,
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
freedom in denominator), where both parameters should be greater than
zero.
The random variate of the F distribution (also known as the
Fisher distribution) is a continuous probability distribution
that arises in ANOVA tests, and is the ratio of two chi-square
variates.
Parameters
----------
dfnum : int or array_like of ints
Degrees of freedom in numerator. Should be greater than zero.
dfden : int or array_like of ints
Degrees of freedom in denominator. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``dfnum`` and ``dfden`` are both scalars.
Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Fisher distribution.
See Also
--------
scipy.stats.f : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The F statistic is used to compare in-group variances to between-group
variances. Calculating the distribution depends on the sampling, and
so it is a function of the respective degrees of freedom in the
problem. The variable `dfnum` is the number of samples minus one, the
between-groups degrees of freedom, while `dfden` is the within-groups
degrees of freedom, the sum of the number of samples in each group
minus the number of groups.
References
----------
.. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
Fifth Edition, 2002.
.. [2] Wikipedia, "F-distribution",
http://en.wikipedia.org/wiki/F-distribution
Examples
--------
An example from Glantz[1], pp 47-40:
Two groups, children of diabetics (25 people) and children from people
without diabetes (25 controls). Fasting blood glucose was measured,
case group had a mean value of 86.1, controls had a mean value of
82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
data consistent with the null hypothesis that the parents diabetic
status does not affect their children's blood glucose levels?
Calculating the F statistic from the data gives a value of 36.01.
Draw samples from the distribution:
>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)
The lower bound for the top 1% of the samples is :
>>> sort(s)[-10]
7.61988120985
So there is about a 1% chance that the F statistic will exceed 7.62,
the measured value is 36, so the null hypothesis is rejected at the 1%
level.
choice(a, size=None, replace=True, p=None)
Generates a random sample from a given 1-D array
.. versionadded:: 1.7.0
Parameters
-----------
a : 1-D array-like or int
If an ndarray, a random sample is generated from its elements.
If an int, the random sample is generated as if a were np.arange(a)
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
replace : boolean, optional
Whether the sample is with or without replacement
p : 1-D array-like, optional
The probabilities associated with each entry in a.
If not given the sample assumes a uniform distribution over all
entries in a.
Returns
--------
samples : single item or ndarray
The generated random samples
Raises
-------
ValueError
If a is an int and less than zero, if a or p are not 1-dimensional,
if a is an array-like of size 0, if p is not a vector of
probabilities, if a and p have different lengths, or if
replace=False and the sample size is greater than the population
size
See Also
---------
randint, shuffle, permutation
Examples
---------
Generate a uniform random sample from np.arange(5) of size 3:
>>> np.random.choice(5, 3)
array([0, 3, 4])
>>> #This is equivalent to np.random.randint(0,5,3)
Generate a non-uniform random sample from np.arange(5) of size 3:
>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0])
Generate a uniform random sample from np.arange(5) of size 3 without
replacement:
>>> np.random.choice(5, 3, replace=False)
array([3,1,0])
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]
Generate a non-uniform random sample from np.arange(5) of size
3 without replacement:
>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0])
Any of the above can be repeated with an arbitrary array-like
instead of just integers. For instance:
>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'],
dtype='|S11')
chisquare(df, size=None)
Draw samples from a chi-square distribution.
When `df` independent random variables, each with standard normal
distributions (mean 0, variance 1), are squared and summed, the
resulting distribution is chi-square (see Notes). This distribution
is often used in hypothesis testing.
Parameters
----------
df : int or array_like of ints
Number of degrees of freedom.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``df`` is a scalar. Otherwise,
``np.array(df).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized chi-square distribution.
Raises
------
ValueError
When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
is given.
Notes
-----
The variable obtained by summing the squares of `df` independent,
standard normally distributed random variables:
.. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i
is chi-square distributed, denoted
.. math:: Q \sim \chi^2_k.
The probability density function of the chi-squared distribution is
.. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
x^{k/2 - 1} e^{-x/2},
where :math:`\Gamma` is the gamma function,
.. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.
References
----------
.. [1] NIST "Engineering Statistics Handbook"
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
Examples
--------
>>> np.random.chisquare(2,4)
array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272])
bytes(length)
Return random bytes.
Parameters
----------
length : int
Number of random bytes.
Returns
-------
out : str
String of length `length`.
Examples
--------
>>> np.random.bytes(10)
' eh\x85\x022SZ\xbf\xa4' #random
binomial(n, p, size=None)
Draw samples from a binomial distribution.
Samples are drawn from a binomial distribution with specified
parameters, n trials and p probability of success where
n an integer >= 0 and p is in the interval [0,1]. (n may be
input as a float, but it is truncated to an integer in use)
Parameters
----------
n : int or array_like of ints
Parameter of the distribution, >= 0. Floats are also accepted,
but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``n`` and ``p`` are both scalars.
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized binomial distribution, where
each sample is equal to the number of successes over the n trials.
See Also
--------
scipy.stats.binom : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the binomial distribution is
.. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N},
where :math:`n` is the number of trials, :math:`p` is the probability
of success, and :math:`N` is the number of successes.
When estimating the standard error of a proportion in a population by
using a random sample, the normal distribution works well unless the
product p*n <=5, where p = population proportion estimate, and n =
number of samples, in which case the binomial distribution is used
instead. For example, a sample of 15 people shows 4 who are left
handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
so the binomial distribution should be used in this case.
References
----------
.. [1] Dalgaard, Peter, "Introductory Statistics with R",
Springer-Verlag, 2002.
.. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
Fifth Edition, 2002.
.. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden
and Quigley, 1972.
.. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/BinomialDistribution.html
.. [5] Wikipedia, "Binomial distribution",
http://en.wikipedia.org/wiki/Binomial_distribution
Examples
--------
Draw samples from the distribution:
>>> n, p = 10, .5 # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.
A real world example. A company drills 9 wild-cat oil exploration
wells, each with an estimated probability of success of 0.1. All nine
wells fail. What is the probability of that happening?
Let's do 20,000 trials of the model, and count the number that
generate zero positive results.
>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.
Unsupported dtype "%s" for randint Seed must be between 0 and 2**32 - 1 RandomState.standard_gamma (line 1810) RandomState.multivariate_normal (line 4369) RandomState.logseries (line 4272) RandomState.lognormal (line 3302) RandomState.hypergeometric (line 4150) RandomState.geometric (line 4082) RandomState.dirichlet (line 4643) RandomState.chisquare (line 2196)
wald(mean, scale, size=None)
Draw samples from a Wald, or inverse Gaussian, distribution.
As the scale approaches infinity, the distribution becomes more like a
Gaussian. Some references claim that the Wald is an inverse Gaussian
with mean equal to 1, but this is by no means universal.
The inverse Gaussian distribution was first studied in relationship to
Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian
because there is an inverse relationship between the time to cover a
unit distance and distance covered in unit time.
Parameters
----------
mean : float or array_like of floats
Distribution mean, should be > 0.
scale : float or array_like of floats
Scale parameter, should be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``mean`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Wald distribution.
Notes
-----
The probability density function for the Wald distribution is
.. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^
\frac{-scale(x-mean)^2}{2\cdotp mean^2x}
As noted above the inverse Gaussian distribution first arise
from attempts to model Brownian motion. It is also a
competitor to the Weibull for use in reliability modeling and
modeling stock returns and interest rate processes.
References
----------
.. [1] Brighton Webs Ltd., Wald Distribution,
http://www.brighton-webs.co.uk/distributions/wald.asp
.. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian
Distribution: Theory : Methodology, and Applications", CRC Press,
1988.
.. [3] Wikipedia, "Wald distribution"
http://en.wikipedia.org/wiki/Wald_distribution
Examples
--------
Draw values from the distribution and plot the histogram:
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True)
>>> plt.show()
standard_normal(size=None)
Draw samples from a standard Normal distribution (mean=0, stdev=1).
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : float or ndarray
Drawn samples.
Examples
--------
>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, #random
-0.38672696, -0.4685006 ]) #random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)
standard_gamma(shape, size=None)
Draw samples from a standard Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters,
shape (sometimes designated "k") and scale=1.
Parameters
----------
shape : float or array_like of floats
Parameter, should be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``shape`` is a scalar. Otherwise,
``np.array(shape).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized standard gamma distribution.
See Also
--------
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gamma distribution is
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
where :math:`k` is the shape and :math:`\theta` the scale,
and :math:`\Gamma` is the Gamma function.
The Gamma distribution is often used to model the times to failure of
electronic components, and arises naturally in processes for which the
waiting times between Poisson distributed events are relevant.
References
----------
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/GammaDistribution.html
.. [2] Wikipedia, "Gamma distribution",
http://en.wikipedia.org/wiki/Gamma_distribution
Examples
--------
Draw samples from the distribution:
>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps
>>> count, bins, ignored = plt.hist(s, 50, normed=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ \
... (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()
standard_exponential(size=None)
Draw samples from the standard exponential distribution.
`standard_exponential` is identical to the exponential distribution
with a scale parameter of 1.
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : float or ndarray
Drawn samples.
Examples
--------
Output a 3x8000 array:
>>> n = np.random.standard_exponential((3, 8000))
standard_cauchy(size=None)
Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
samples : ndarray or scalar
The drawn samples.
Notes
-----
The probability density function for the full Cauchy distribution is
.. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
(\frac{x-x_0}{\gamma})^2 \bigr] }
and the Standard Cauchy distribution just sets :math:`x_0=0` and
:math:`\gamma=1`
The Cauchy distribution arises in the solution to the driven harmonic
oscillator problem, and also describes spectral line broadening. It
also describes the distribution of values at which a line tilted at
a random angle will cut the x axis.
When studying hypothesis tests that assume normality, seeing how the
tests perform on data from a Cauchy distribution is a good indicator of
their sensitivity to a heavy-tailed distribution, since the Cauchy looks
very much like a Gaussian distribution, but with heavier tails.
References
----------
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy
Distribution",
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
.. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/CauchyDistribution.html
.. [3] Wikipedia, "Cauchy distribution"
http://en.wikipedia.org/wiki/Cauchy_distribution
Examples
--------
Draw samples and plot the distribution:
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()
shuffle(x)
Modify a sequence in-place by shuffling its contents.
This function only shuffles the array along the first axis of a
multi-dimensional array. The order of sub-arrays is changed but
their contents remains the same.
Parameters
----------
x : array_like
The array or list to be shuffled.
Returns
-------
None
Examples
--------
>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8]
Multi-dimensional arrays are only shuffled along the first axis:
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5],
[6, 7, 8],
[0, 1, 2]])
random_sample(size=None)
Return random floats in the half-open interval [0.0, 1.0).
Results are from the "continuous uniform" distribution over the
stated interval. To sample :math:`Unif[a, b), b > a` multiply
the output of `random_sample` by `(b-a)` and add `a`::
(b - a) * random_sample() + a
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : float or ndarray of floats
Array of random floats of shape `size` (unless ``size=None``, in which
case a single float is returned).
Examples
--------
>>> np.random.random_sample()
0.47108547995356098
>>> type(np.random.random_sample())
<type 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428])
Three-by-two array of random numbers from [-5, 0):
>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984],
[-2.99091858, -0.79479508],
[-1.23204345, -1.75224494]])
randn(d0, d1, ..., dn)
Return a sample (or samples) from the "standard normal" distribution.
If positive, int_like or int-convertible arguments are provided,
`randn` generates an array of shape ``(d0, d1, ..., dn)``, filled
with random floats sampled from a univariate "normal" (Gaussian)
distribution of mean 0 and variance 1 (if any of the :math:`d_i` are
floats, they are first converted to integers by truncation). A single
float randomly sampled from the distribution is returned if no
argument is provided.
This is a convenience function. If you want an interface that takes a
tuple as the first argument, use `numpy.random.standard_normal` instead.
Parameters
----------
d0, d1, ..., dn : int, optional
The dimensions of the returned array, should be all positive.
If no argument is given a single Python float is returned.
Returns
-------
Z : ndarray or float
A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from
the standard normal distribution, or a single such float if
no parameters were supplied.
See Also
--------
random.standard_normal : Similar, but takes a tuple as its argument.
Notes
-----
For random samples from :math:`N(\mu, \sigma^2)`, use:
``sigma * np.random.randn(...) + mu``
Examples
--------
>>> np.random.randn()
2.1923875335537315 #random
Two-by-four array of samples from N(3, 6.25):
>>> 2.5 * np.random.randn(2, 4) + 3
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], #random
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) #random
rand(d0, d1, ..., dn)
Random values in a given shape.
Create an array of the given shape and populate it with
random samples from a uniform distribution
over ``[0, 1)``.
Parameters
----------
d0, d1, ..., dn : int, optional
The dimensions of the returned array, should all be positive.
If no argument is given a single Python float is returned.
Returns
-------
out : ndarray, shape ``(d0, d1, ..., dn)``
Random values.
See Also
--------
random
Notes
-----
This is a convenience function. If you want an interface that
takes a shape-tuple as the first argument, refer to
np.random.random_sample .
Examples
--------
>>> np.random.rand(3,2)
array([[ 0.14022471, 0.96360618], #random
[ 0.37601032, 0.25528411], #random
[ 0.49313049, 0.94909878]]) #random
poisson(lam=1.0, size=None)
Draw samples from a Poisson distribution.
The Poisson distribution is the limit of the binomial distribution
for large N.
Parameters
----------
lam : float or array_like of floats
Expectation of interval, should be >= 0. A sequence of expectation
intervals must be broadcastable over the requested size.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``lam`` is a scalar. Otherwise,
``np.array(lam).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Poisson distribution.
Notes
-----
The Poisson distribution
.. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}
For events with an expected separation :math:`\lambda` the Poisson
distribution :math:`f(k; \lambda)` describes the probability of
:math:`k` events occurring within the observed
interval :math:`\lambda`.
Because the output is limited to the range of the C long type, a
ValueError is raised when `lam` is within 10 sigma of the maximum
representable value.
References
----------
.. [1] Weisstein, Eric W. "Poisson Distribution."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/PoissonDistribution.html
.. [2] Wikipedia, "Poisson distribution",
http://en.wikipedia.org/wiki/Poisson_distribution
Examples
--------
Draw samples from the distribution:
>>> import numpy as np
>>> s = np.random.poisson(5, 10000)
Display histogram of the sample:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, normed=True)
>>> plt.show()
Draw each 100 values for lambda 100 and 500:
>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
permutation(x)
Randomly permute a sequence, or return a permuted range.
If `x` is a multi-dimensional array, it is only shuffled along its
first index.
Parameters
----------
x : int or array_like
If `x` is an integer, randomly permute ``np.arange(x)``.
If `x` is an array, make a copy and shuffle the elements
randomly.
Returns
-------
out : ndarray
Permuted sequence or array range.
Examples
--------
>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6])
>>> np.random.permutation([1, 4, 9, 12, 15])
array([15, 1, 9, 4, 12])
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8],
[0, 1, 2],
[3, 4, 5]])
laplace(loc=0.0, scale=1.0, size=None)
Draw samples from the Laplace or double exponential distribution with
specified location (or mean) and scale (decay).
The Laplace distribution is similar to the Gaussian/normal distribution,
but is sharper at the peak and has fatter tails. It represents the
difference between two independent, identically distributed exponential
random variables.
Parameters
----------
loc : float or array_like of floats, optional
The position, :math:`\mu`, of the distribution peak. Default is 0.
scale : float or array_like of floats, optional
:math:`\lambda`, the exponential decay. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Laplace distribution.
Notes
-----
It has the probability density function
.. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda}
\exp\left(-\frac{|x - \mu|}{\lambda}\right).
The first law of Laplace, from 1774, states that the frequency
of an error can be expressed as an exponential function of the
absolute magnitude of the error, which leads to the Laplace
distribution. For many problems in economics and health
sciences, this distribution seems to model the data better
than the standard Gaussian distribution.
References
----------
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing," New York: Dover, 1972.
.. [2] Kotz, Samuel, et. al. "The Laplace Distribution and
Generalizations, " Birkhauser, 2001.
.. [3] Weisstein, Eric W. "Laplace Distribution."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/LaplaceDistribution.html
.. [4] Wikipedia, "Laplace distribution",
http://en.wikipedia.org/wiki/Laplace_distribution
Examples
--------
Draw samples from the distribution
>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)
Plot Gaussian for comparison:
>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
... np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)
dirichlet(alpha, size=None)
Draw samples from the Dirichlet distribution.
Draw `size` samples of dimension k from a Dirichlet distribution. A
Dirichlet-distributed random variable can be seen as a multivariate
generalization of a Beta distribution. Dirichlet pdf is the conjugate
prior of a multinomial in Bayesian inference.
Parameters
----------
alpha : array
Parameter of the distribution (k dimension for sample of
dimension k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
samples : ndarray,
The drawn samples, of shape (size, alpha.ndim).
Notes
-----
.. math:: X \approx \prod_{i=1}^{k}{x^{\alpha_i-1}_i}
Uses the following property for computation: for each dimension,
draw a random sample y_i from a standard gamma generator of shape
`alpha_i`, then
:math:`X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)` is
Dirichlet distributed.
References
----------
.. [1] David McKay, "Information Theory, Inference and Learning
Algorithms," chapter 23,
http://www.inference.phy.cam.ac.uk/mackay/
.. [2] Wikipedia, "Dirichlet distribution",
http://en.wikipedia.org/wiki/Dirichlet_distribution
Examples
--------
Taking an example cited in Wikipedia, this distribution can be used if
one wanted to cut strings (each of initial length 1.0) into K pieces
with different lengths, where each piece had, on average, a designated
average length, but allowing some variation in the relative sizes of
the pieces.
>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
RandomState.vonmises (line 2551) RandomState.rayleigh (line 3426) RandomState.logistic (line 3209) RandomState.binomial (line 3686) probabilities do not sum to 1 RandomState.weibull (line 2759) RandomState.uniform (line 1210) RandomState.tomaxint (line 858) RandomState.shuffle (line 4759) RandomState.poisson (line 3903) RandomState.laplace (line 2980) RandomState.randint (line 905) RandomState.pareto (line 2649) RandomState.normal (line 1547) RandomState.gumbel (line 3078) RandomState.choice (line 1028) high is out of bounds for %s a and p must have same size RandomState.randn (line 1360) RandomState.power (line 2869) RandomState.gamma (line 1896) mean must be 1 dimensional low is out of bounds for %s Range exceeds valid bounds RandomState.zipf (line 3991) RandomState.wald (line 3505) RandomState.rand (line 1316) RandomState.bytes (line 999) algorithm must be 'MT19937' a must be greater than 0 state must be 624 longs p must be 1-dimensional a must be 1-dimensional RandomState.f (line 1992) lam value too large. standard_exponential noncentral_chisquare multivariate_normal lam value too large a must be non-empty ngood + nbad < nsample DeprecationWarning negative_binomial __RandomState_ctor sum(pvals[:-1]) > 1.0 standard_normal standard_cauchy _shape_from_size random_integers poisson_lam_max dummy_threading standard_gamma hypergeometric RuntimeWarning random_sample greater_equal count_nonzero OverflowError searchsorted return_index _randint_type noncentral_f scale <= 0.0 _rand_uint64 _rand_uint32 _rand_uint16 permutation multinomial exponential check_valid ImportError triangular standard_t _rand_uint8 _rand_int64 _rand_int32 _rand_int16 __pyx_vtable__ numpy.dual mode > right logical_or less_equal left == right issubdtype empty_like array_data ValueError threading sigma < 0.0 set_state scale <= 0 scale < 0.0 _rand_int8 _rand_bool nsample < 1 logseries lognormal left > mode get_state geometric dirichlet chisquare broadcast TypeError warnings vonmises subtract rngstate reversed rayleigh p is nan operator mean <= 0.0 low >= high logistic itemsize isfinite floating binomial allclose weibull uniform strides signbit sigma < 0 shuffle shape < 0 scale < 0 reshape replace randint poisson p >= 1.0 p <= 0.0 nsample ngood < 0 ndarray laplace kappa < 0 integer greater float64 dfnum <= 1 dfnum <= 0 dfden <= 0 casting asarray MT19937 unsafe unique uint64 uint32 uint16 reduce random _rand pareto normal nonc < 0 nbad < 0 mtrand mean <= 0 __import__ ignore gumbel format cumsum ctypes choice bool astype arange zeros uint8 state sigma shape scale right ravel range randn raise pvals power p > 1.0 p < 0.0 numpy ngood n <= 0 lam < 0 kappa isnan int64 int32 int16 index iinfo gamma finfo equal __enter__ empty dtype dfnum dfden bytes array alpha a <= 1.0 a < 0 zipf warn wald uint __test__ take sqrt sort size side seed safe rtol rand prod nonc ndim nbad name mode mean __main__ long less left item intp int8 high __exit__ df <= 0 data copy bool_ beta atol Lock tol svd rng p > 1 p < 0 out off n < 0 max low loc lam int eps dot cov cnt buf b <= 0 any all add a <= 0 np mu df p n l f d b a T L ð¿ ð? ð?:0âyE>ÿÿÿÿÿÿÿ rb /dev/random /dev/urandom no error random device unvavailable A < À UUUUUUµ?lÁlÁf¿ J?88C¿$ÿ+ K?<ٰj_¿¤A¤Az?SˆB¿
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permutation(x)
Randomly permute a sequence, or return a permuted range.
If `x` is a multi-dimensional array, it is only shuffled along its
first index.
Parameters
----------
x : int or array_like
If `x` is an integer, randomly permute ``np.arange(x)``.
If `x` is an array, make a copy and shuffle the elements
randomly.
Returns
-------
out : ndarray
Permuted sequence or array range.
Examples
--------
>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6])
>>> np.random.permutation([1, 4, 9, 12, 15])
array([15, 1, 9, 4, 12])
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8],
[0, 1, 2],
[3, 4, 5]])
shuffle(x)
Modify a sequence in-place by shuffling its contents.
This function only shuffles the array along the first axis of a
multi-dimensional array. The order of sub-arrays is changed but
their contents remains the same.
Parameters
----------
x : array_like
The array or list to be shuffled.
Returns
-------
None
Examples
--------
>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8]
Multi-dimensional arrays are only shuffled along the first axis:
>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5],
[6, 7, 8],
[0, 1, 2]])
dirichlet(alpha, size=None)
Draw samples from the Dirichlet distribution.
Draw `size` samples of dimension k from a Dirichlet distribution. A
Dirichlet-distributed random variable can be seen as a multivariate
generalization of a Beta distribution. Dirichlet pdf is the conjugate
prior of a multinomial in Bayesian inference.
Parameters
----------
alpha : array
Parameter of the distribution (k dimension for sample of
dimension k).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
samples : ndarray,
The drawn samples, of shape (size, alpha.ndim).
Notes
-----
.. math:: X \approx \prod_{i=1}^{k}{x^{\alpha_i-1}_i}
Uses the following property for computation: for each dimension,
draw a random sample y_i from a standard gamma generator of shape
`alpha_i`, then
:math:`X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)` is
Dirichlet distributed.
References
----------
.. [1] David McKay, "Information Theory, Inference and Learning
Algorithms," chapter 23,
http://www.inference.phy.cam.ac.uk/mackay/
.. [2] Wikipedia, "Dirichlet distribution",
http://en.wikipedia.org/wiki/Dirichlet_distribution
Examples
--------
Taking an example cited in Wikipedia, this distribution can be used if
one wanted to cut strings (each of initial length 1.0) into K pieces
with different lengths, where each piece had, on average, a designated
average length, but allowing some variation in the relative sizes of
the pieces.
>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
multinomial(n, pvals, size=None)
Draw samples from a multinomial distribution.
The multinomial distribution is a multivariate generalisation of the
binomial distribution. Take an experiment with one of ``p``
possible outcomes. An example of such an experiment is throwing a dice,
where the outcome can be 1 through 6. Each sample drawn from the
distribution represents `n` such experiments. Its values,
``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the
outcome was ``i``.
Parameters
----------
n : int
Number of experiments.
pvals : sequence of floats, length p
Probabilities of each of the ``p`` different outcomes. These
should sum to 1 (however, the last element is always assumed to
account for the remaining probability, as long as
``sum(pvals[:-1]) <= 1)``.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : ndarray
The drawn samples, of shape *size*, if that was provided. If not,
the shape is ``(N,)``.
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
value drawn from the distribution.
Examples
--------
Throw a dice 20 times:
>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]])
It landed 4 times on 1, once on 2, etc.
Now, throw the dice 20 times, and 20 times again:
>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3],
[2, 4, 3, 4, 0, 7]])
For the first run, we threw 3 times 1, 4 times 2, etc. For the second,
we threw 2 times 1, 4 times 2, etc.
A loaded die is more likely to land on number 6:
>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26])
The probability inputs should be normalized. As an implementation
detail, the value of the last entry is ignored and assumed to take
up any leftover probability mass, but this should not be relied on.
A biased coin which has twice as much weight on one side as on the
other should be sampled like so:
>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT
array([38, 62])
not like:
>>> np.random.multinomial(100, [1.0, 2.0]) # WRONG
array([100, 0])
multivariate_normal(mean, cov[, size, check_valid, tol])
Draw random samples from a multivariate normal distribution.
The multivariate normal, multinormal or Gaussian distribution is a
generalization of the one-dimensional normal distribution to higher
dimensions. Such a distribution is specified by its mean and
covariance matrix. These parameters are analogous to the mean
(average or "center") and variance (standard deviation, or "width,"
squared) of the one-dimensional normal distribution.
Parameters
----------
mean : 1-D array_like, of length N
Mean of the N-dimensional distribution.
cov : 2-D array_like, of shape (N, N)
Covariance matrix of the distribution. It must be symmetric and
positive-semidefinite for proper sampling.
size : int or tuple of ints, optional
Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because
each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
If no shape is specified, a single (`N`-D) sample is returned.
check_valid : { 'warn', 'raise', 'ignore' }, optional
Behavior when the covariance matrix is not positive semidefinite.
tol : float, optional
Tolerance when checking the singular values in covariance matrix.
Returns
-------
out : ndarray
The drawn samples, of shape *size*, if that was provided. If not,
the shape is ``(N,)``.
In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
value drawn from the distribution.
Notes
-----
The mean is a coordinate in N-dimensional space, which represents the
location where samples are most likely to be generated. This is
analogous to the peak of the bell curve for the one-dimensional or
univariate normal distribution.
Covariance indicates the level to which two variables vary together.
From the multivariate normal distribution, we draw N-dimensional
samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix
element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
"spread").
Instead of specifying the full covariance matrix, popular
approximations include:
- Spherical covariance (`cov` is a multiple of the identity matrix)
- Diagonal covariance (`cov` has non-negative elements, and only on
the diagonal)
This geometrical property can be seen in two dimensions by plotting
generated data-points:
>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]] # diagonal covariance
Diagonal covariance means that points are oriented along x or y-axis:
>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()
Note that the covariance matrix must be positive semidefinite (a.k.a.
nonnegative-definite). Otherwise, the behavior of this method is
undefined and backwards compatibility is not guaranteed.
References
----------
.. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
Processes," 3rd ed., New York: McGraw-Hill, 1991.
.. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
Classification," 2nd ed., New York: Wiley, 2001.
Examples
--------
>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)
The following is probably true, given that 0.6 is roughly twice the
standard deviation:
>>> list((x[0,0,:] - mean) < 0.6)
[True, True]
logseries(p, size=None)
Draw samples from a logarithmic series distribution.
Samples are drawn from a log series distribution with specified
shape parameter, 0 < ``p`` < 1.
Parameters
----------
p : float or array_like of floats
Shape parameter for the distribution. Must be in the range (0, 1).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``p`` is a scalar. Otherwise,
``np.array(p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized logarithmic series distribution.
See Also
--------
scipy.stats.logser : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Log Series distribution is
.. math:: P(k) = \frac{-p^k}{k \ln(1-p)},
where p = probability.
The log series distribution is frequently used to represent species
richness and occurrence, first proposed by Fisher, Corbet, and
Williams in 1943 [2]. It may also be used to model the numbers of
occupants seen in cars [3].
References
----------
.. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional
species diversity through the log series distribution of
occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
Volume 5, Number 5, September 1999 , pp. 187-195(9).
.. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
relation between the number of species and the number of
individuals in a random sample of an animal population.
Journal of Animal Ecology, 12:42-58.
.. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
Data Sets, CRC Press, 1994.
.. [4] Wikipedia, "Logarithmic distribution",
http://en.wikipedia.org/wiki/Logarithmic_distribution
Examples
--------
Draw samples from the distribution:
>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> count, bins, ignored = plt.hist(s)
# plot against distribution
>>> def logseries(k, p):
... return -p**k/(k*log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
logseries(bins, a).max(), 'r')
>>> plt.show()
hypergeometric(ngood, nbad, nsample, size=None)
Draw samples from a Hypergeometric distribution.
Samples are drawn from a hypergeometric distribution with specified
parameters, ngood (ways to make a good selection), nbad (ways to make
a bad selection), and nsample = number of items sampled, which is less
than or equal to the sum ngood + nbad.
Parameters
----------
ngood : int or array_like of ints
Number of ways to make a good selection. Must be nonnegative.
nbad : int or array_like of ints
Number of ways to make a bad selection. Must be nonnegative.
nsample : int or array_like of ints
Number of items sampled. Must be at least 1 and at most
``ngood + nbad``.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``ngood``, ``nbad``, and ``nsample``
are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size``
samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized hypergeometric distribution.
See Also
--------
scipy.stats.hypergeom : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Hypergeometric distribution is
.. math:: P(x) = \frac{\binom{m}{n}\binom{N-m}{n-x}}{\binom{N}{n}},
where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n`
for P(x) the probability of x successes, n = ngood, m = nbad, and
N = number of samples.
Consider an urn with black and white marbles in it, ngood of them
black and nbad are white. If you draw nsample balls without
replacement, then the hypergeometric distribution describes the
distribution of black balls in the drawn sample.
Note that this distribution is very similar to the binomial
distribution, except that in this case, samples are drawn without
replacement, whereas in the Binomial case samples are drawn with
replacement (or the sample space is infinite). As the sample space
becomes large, this distribution approaches the binomial.
References
----------
.. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
and Quigley, 1972.
.. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/HypergeometricDistribution.html
.. [3] Wikipedia, "Hypergeometric distribution",
http://en.wikipedia.org/wiki/Hypergeometric_distribution
Examples
--------
Draw samples from the distribution:
>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> hist(s)
# note that it is very unlikely to grab both bad items
Suppose you have an urn with 15 white and 15 black marbles.
If you pull 15 marbles at random, how likely is it that
12 or more of them are one color?
>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
# answer = 0.003 ... pretty unlikely!
geometric(p, size=None)
Draw samples from the geometric distribution.
Bernoulli trials are experiments with one of two outcomes:
success or failure (an example of such an experiment is flipping
a coin). The geometric distribution models the number of trials
that must be run in order to achieve success. It is therefore
supported on the positive integers, ``k = 1, 2, ...``.
The probability mass function of the geometric distribution is
.. math:: f(k) = (1 - p)^{k - 1} p
where `p` is the probability of success of an individual trial.
Parameters
----------
p : float or array_like of floats
The probability of success of an individual trial.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``p`` is a scalar. Otherwise,
``np.array(p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized geometric distribution.
Examples
--------
Draw ten thousand values from the geometric distribution,
with the probability of an individual success equal to 0.35:
>>> z = np.random.geometric(p=0.35, size=10000)
How many trials succeeded after a single run?
>>> (z == 1).sum() / 10000.
0.34889999999999999 #random
zipf(a, size=None)
Draw samples from a Zipf distribution.
Samples are drawn from a Zipf distribution with specified parameter
`a` > 1.
The Zipf distribution (also known as the zeta distribution) is a
continuous probability distribution that satisfies Zipf's law: the
frequency of an item is inversely proportional to its rank in a
frequency table.
Parameters
----------
a : float or array_like of floats
Distribution parameter. Should be greater than 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Zipf distribution.
See Also
--------
scipy.stats.zipf : probability density function, distribution, or
cumulative density function, etc.
Notes
-----
The probability density for the Zipf distribution is
.. math:: p(x) = \frac{x^{-a}}{\zeta(a)},
where :math:`\zeta` is the Riemann Zeta function.
It is named for the American linguist George Kingsley Zipf, who noted
that the frequency of any word in a sample of a language is inversely
proportional to its rank in the frequency table.
References
----------
.. [1] Zipf, G. K., "Selected Studies of the Principle of Relative
Frequency in Language," Cambridge, MA: Harvard Univ. Press,
1932.
Examples
--------
Draw samples from the distribution:
>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> from scipy import special
Truncate s values at 50 so plot is interesting:
>>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)
>>> plt.plot(x, y/max(y), linewidth=2, color='r')
>>> plt.show()
poisson(lam=1.0, size=None)
Draw samples from a Poisson distribution.
The Poisson distribution is the limit of the binomial distribution
for large N.
Parameters
----------
lam : float or array_like of floats
Expectation of interval, should be >= 0. A sequence of expectation
intervals must be broadcastable over the requested size.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``lam`` is a scalar. Otherwise,
``np.array(lam).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Poisson distribution.
Notes
-----
The Poisson distribution
.. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}
For events with an expected separation :math:`\lambda` the Poisson
distribution :math:`f(k; \lambda)` describes the probability of
:math:`k` events occurring within the observed
interval :math:`\lambda`.
Because the output is limited to the range of the C long type, a
ValueError is raised when `lam` is within 10 sigma of the maximum
representable value.
References
----------
.. [1] Weisstein, Eric W. "Poisson Distribution."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/PoissonDistribution.html
.. [2] Wikipedia, "Poisson distribution",
http://en.wikipedia.org/wiki/Poisson_distribution
Examples
--------
Draw samples from the distribution:
>>> import numpy as np
>>> s = np.random.poisson(5, 10000)
Display histogram of the sample:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, normed=True)
>>> plt.show()
Draw each 100 values for lambda 100 and 500:
>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))
negative_binomial(n, p, size=None)
Draw samples from a negative binomial distribution.
Samples are drawn from a negative binomial distribution with specified
parameters, `n` trials and `p` probability of success where `n` is an
integer > 0 and `p` is in the interval [0, 1].
Parameters
----------
n : int or array_like of ints
Parameter of the distribution, > 0. Floats are also accepted,
but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``n`` and ``p`` are both scalars.
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized negative binomial distribution,
where each sample is equal to N, the number of trials it took to
achieve n - 1 successes, N - (n - 1) failures, and a success on the,
(N + n)th trial.
Notes
-----
The probability density for the negative binomial distribution is
.. math:: P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N},
where :math:`n-1` is the number of successes, :math:`p` is the
probability of success, and :math:`N+n-1` is the number of trials.
The negative binomial distribution gives the probability of n-1
successes and N failures in N+n-1 trials, and success on the (N+n)th
trial.
If one throws a die repeatedly until the third time a "1" appears,
then the probability distribution of the number of non-"1"s that
appear before the third "1" is a negative binomial distribution.
References
----------
.. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/NegativeBinomialDistribution.html
.. [2] Wikipedia, "Negative binomial distribution",
http://en.wikipedia.org/wiki/Negative_binomial_distribution
Examples
--------
Draw samples from the distribution:
A real world example. A company drills wild-cat oil
exploration wells, each with an estimated probability of
success of 0.1. What is the probability of having one success
for each successive well, that is what is the probability of a
single success after drilling 5 wells, after 6 wells, etc.?
>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11):
... probability = sum(s<i) / 100000.
... print i, "wells drilled, probability of one success =", probability
binomial(n, p, size=None)
Draw samples from a binomial distribution.
Samples are drawn from a binomial distribution with specified
parameters, n trials and p probability of success where
n an integer >= 0 and p is in the interval [0,1]. (n may be
input as a float, but it is truncated to an integer in use)
Parameters
----------
n : int or array_like of ints
Parameter of the distribution, >= 0. Floats are also accepted,
but they will be truncated to integers.
p : float or array_like of floats
Parameter of the distribution, >= 0 and <=1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``n`` and ``p`` are both scalars.
Otherwise, ``np.broadcast(n, p).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized binomial distribution, where
each sample is equal to the number of successes over the n trials.
See Also
--------
scipy.stats.binom : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the binomial distribution is
.. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N},
where :math:`n` is the number of trials, :math:`p` is the probability
of success, and :math:`N` is the number of successes.
When estimating the standard error of a proportion in a population by
using a random sample, the normal distribution works well unless the
product p*n <=5, where p = population proportion estimate, and n =
number of samples, in which case the binomial distribution is used
instead. For example, a sample of 15 people shows 4 who are left
handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
so the binomial distribution should be used in this case.
References
----------
.. [1] Dalgaard, Peter, "Introductory Statistics with R",
Springer-Verlag, 2002.
.. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
Fifth Edition, 2002.
.. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden
and Quigley, 1972.
.. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/BinomialDistribution.html
.. [5] Wikipedia, "Binomial distribution",
http://en.wikipedia.org/wiki/Binomial_distribution
Examples
--------
Draw samples from the distribution:
>>> n, p = 10, .5 # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.
A real world example. A company drills 9 wild-cat oil exploration
wells, each with an estimated probability of success of 0.1. All nine
wells fail. What is the probability of that happening?
Let's do 20,000 trials of the model, and count the number that
generate zero positive results.
>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.
triangular(left, mode, right, size=None)
Draw samples from the triangular distribution over the
interval ``[left, right]``.
The triangular distribution is a continuous probability
distribution with lower limit left, peak at mode, and upper
limit right. Unlike the other distributions, these parameters
directly define the shape of the pdf.
Parameters
----------
left : float or array_like of floats
Lower limit.
mode : float or array_like of floats
The value where the peak of the distribution occurs.
The value should fulfill the condition ``left <= mode <= right``.
right : float or array_like of floats
Upper limit, should be larger than `left`.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``left``, ``mode``, and ``right``
are all scalars. Otherwise, ``np.broadcast(left, mode, right).size``
samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized triangular distribution.
Notes
-----
The probability density function for the triangular distribution is
.. math:: P(x;l, m, r) = \begin{cases}
\frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\
\frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\
0& \text{otherwise}.
\end{cases}
The triangular distribution is often used in ill-defined
problems where the underlying distribution is not known, but
some knowledge of the limits and mode exists. Often it is used
in simulations.
References
----------
.. [1] Wikipedia, "Triangular distribution"
http://en.wikipedia.org/wiki/Triangular_distribution
Examples
--------
Draw values from the distribution and plot the histogram:
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
... normed=True)
>>> plt.show()
wald(mean, scale, size=None)
Draw samples from a Wald, or inverse Gaussian, distribution.
As the scale approaches infinity, the distribution becomes more like a
Gaussian. Some references claim that the Wald is an inverse Gaussian
with mean equal to 1, but this is by no means universal.
The inverse Gaussian distribution was first studied in relationship to
Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian
because there is an inverse relationship between the time to cover a
unit distance and distance covered in unit time.
Parameters
----------
mean : float or array_like of floats
Distribution mean, should be > 0.
scale : float or array_like of floats
Scale parameter, should be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``mean`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Wald distribution.
Notes
-----
The probability density function for the Wald distribution is
.. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^
\frac{-scale(x-mean)^2}{2\cdotp mean^2x}
As noted above the inverse Gaussian distribution first arise
from attempts to model Brownian motion. It is also a
competitor to the Weibull for use in reliability modeling and
modeling stock returns and interest rate processes.
References
----------
.. [1] Brighton Webs Ltd., Wald Distribution,
http://www.brighton-webs.co.uk/distributions/wald.asp
.. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian
Distribution: Theory : Methodology, and Applications", CRC Press,
1988.
.. [3] Wikipedia, "Wald distribution"
http://en.wikipedia.org/wiki/Wald_distribution
Examples
--------
Draw values from the distribution and plot the histogram:
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, normed=True)
>>> plt.show()
rayleigh(scale=1.0, size=None)
Draw samples from a Rayleigh distribution.
The :math:`\chi` and Weibull distributions are generalizations of the
Rayleigh.
Parameters
----------
scale : float or array_like of floats, optional
Scale, also equals the mode. Should be >= 0. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``scale`` is a scalar. Otherwise,
``np.array(scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Rayleigh distribution.
Notes
-----
The probability density function for the Rayleigh distribution is
.. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}
The Rayleigh distribution would arise, for example, if the East
and North components of the wind velocity had identical zero-mean
Gaussian distributions. Then the wind speed would have a Rayleigh
distribution.
References
----------
.. [1] Brighton Webs Ltd., "Rayleigh Distribution,"
http://www.brighton-webs.co.uk/distributions/rayleigh.asp
.. [2] Wikipedia, "Rayleigh distribution"
http://en.wikipedia.org/wiki/Rayleigh_distribution
Examples
--------
Draw values from the distribution and plot the histogram
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, normed=True)
Wave heights tend to follow a Rayleigh distribution. If the mean wave
height is 1 meter, what fraction of waves are likely to be larger than 3
meters?
>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)
The percentage of waves larger than 3 meters is:
>>> 100.*sum(s>3)/1000000.
0.087300000000000003
lognormal(mean=0.0, sigma=1.0, size=None)
Draw samples from a log-normal distribution.
Draw samples from a log-normal distribution with specified mean,
standard deviation, and array shape. Note that the mean and standard
deviation are not the values for the distribution itself, but of the
underlying normal distribution it is derived from.
Parameters
----------
mean : float or array_like of floats, optional
Mean value of the underlying normal distribution. Default is 0.
sigma : float or array_like of floats, optional
Standard deviation of the underlying normal distribution. Should
be greater than zero. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``mean`` and ``sigma`` are both scalars.
Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized log-normal distribution.
See Also
--------
scipy.stats.lognorm : probability density function, distribution,
cumulative density function, etc.
Notes
-----
A variable `x` has a log-normal distribution if `log(x)` is normally
distributed. The probability density function for the log-normal
distribution is:
.. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}
where :math:`\mu` is the mean and :math:`\sigma` is the standard
deviation of the normally distributed logarithm of the variable.
A log-normal distribution results if a random variable is the *product*
of a large number of independent, identically-distributed variables in
the same way that a normal distribution results if the variable is the
*sum* of a large number of independent, identically-distributed
variables.
References
----------
.. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal
Distributions across the Sciences: Keys and Clues,"
BioScience, Vol. 51, No. 5, May, 2001.
http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
.. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme
Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, normed=True, align='mid')
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
... / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()
Demonstrate that taking the products of random samples from a uniform
distribution can be fit well by a log-normal probability density
function.
>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
... a = 10. + np.random.random(100)
... b.append(np.product(a))
>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, normed=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
... / (x * sigma * np.sqrt(2 * np.pi)))
>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()
logistic(loc=0.0, scale=1.0, size=None)
Draw samples from a logistic distribution.
Samples are drawn from a logistic distribution with specified
parameters, loc (location or mean, also median), and scale (>0).
Parameters
----------
loc : float or array_like of floats, optional
Parameter of the distribution. Default is 0.
scale : float or array_like of floats, optional
Parameter of the distribution. Should be greater than zero.
Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized logistic distribution.
See Also
--------
scipy.stats.logistic : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Logistic distribution is
.. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},
where :math:`\mu` = location and :math:`s` = scale.
The Logistic distribution is used in Extreme Value problems where it
can act as a mixture of Gumbel distributions, in Epidemiology, and by
the World Chess Federation (FIDE) where it is used in the Elo ranking
system, assuming the performance of each player is a logistically
distributed random variable.
References
----------
.. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of
Extreme Values, from Insurance, Finance, Hydrology and Other
Fields," Birkhauser Verlag, Basel, pp 132-133.
.. [2] Weisstein, Eric W. "Logistic Distribution." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/LogisticDistribution.html
.. [3] Wikipedia, "Logistic-distribution",
http://en.wikipedia.org/wiki/Logistic_distribution
Examples
--------
Draw samples from the distribution:
>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> count, bins, ignored = plt.hist(s, bins=50)
# plot against distribution
>>> def logist(x, loc, scale):
... return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2)
>>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\
... logist(bins, loc, scale).max())
>>> plt.show()
gumbel(loc=0.0, scale=1.0, size=None)
Draw samples from a Gumbel distribution.
Draw samples from a Gumbel distribution with specified location and
scale. For more information on the Gumbel distribution, see
Notes and References below.
Parameters
----------
loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Gumbel distribution.
See Also
--------
scipy.stats.gumbel_l
scipy.stats.gumbel_r
scipy.stats.genextreme
weibull
Notes
-----
The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
Value Type I) distribution is one of a class of Generalized Extreme
Value (GEV) distributions used in modeling extreme value problems.
The Gumbel is a special case of the Extreme Value Type I distribution
for maximums from distributions with "exponential-like" tails.
The probability density for the Gumbel distribution is
.. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
\beta}},
where :math:`\mu` is the mode, a location parameter, and
:math:`\beta` is the scale parameter.
The Gumbel (named for German mathematician Emil Julius Gumbel) was used
very early in the hydrology literature, for modeling the occurrence of
flood events. It is also used for modeling maximum wind speed and
rainfall rates. It is a "fat-tailed" distribution - the probability of
an event in the tail of the distribution is larger than if one used a
Gaussian, hence the surprisingly frequent occurrence of 100-year
floods. Floods were initially modeled as a Gaussian process, which
underestimated the frequency of extreme events.
It is one of a class of extreme value distributions, the Generalized
Extreme Value (GEV) distributions, which also includes the Weibull and
Frechet.
The function has a mean of :math:`\mu + 0.57721\beta` and a variance
of :math:`\frac{\pi^2}{6}\beta^2`.
References
----------
.. [1] Gumbel, E. J., "Statistics of Extremes,"
New York: Columbia University Press, 1958.
.. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
Values from Insurance, Finance, Hydrology and Other Fields,"
Basel: Birkhauser Verlag, 2001.
Examples
--------
Draw samples from the distribution:
>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
... * np.exp( -np.exp( -(bins - mu) /beta) ),
... linewidth=2, color='r')
>>> plt.show()
Show how an extreme value distribution can arise from a Gaussian process
and compare to a Gaussian:
>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
... a = np.random.normal(mu, beta, 1000)
... means.append(a.mean())
... maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, normed=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
... * np.exp(-np.exp(-(bins - mu)/beta)),
... linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
... * np.exp(-(bins - mu)**2 / (2 * beta**2)),
... linewidth=2, color='g')
>>> plt.show()
laplace(loc=0.0, scale=1.0, size=None)
Draw samples from the Laplace or double exponential distribution with
specified location (or mean) and scale (decay).
The Laplace distribution is similar to the Gaussian/normal distribution,
but is sharper at the peak and has fatter tails. It represents the
difference between two independent, identically distributed exponential
random variables.
Parameters
----------
loc : float or array_like of floats, optional
The position, :math:`\mu`, of the distribution peak. Default is 0.
scale : float or array_like of floats, optional
:math:`\lambda`, the exponential decay. Default is 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Laplace distribution.
Notes
-----
It has the probability density function
.. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda}
\exp\left(-\frac{|x - \mu|}{\lambda}\right).
The first law of Laplace, from 1774, states that the frequency
of an error can be expressed as an exponential function of the
absolute magnitude of the error, which leads to the Laplace
distribution. For many problems in economics and health
sciences, this distribution seems to model the data better
than the standard Gaussian distribution.
References
----------
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing," New York: Dover, 1972.
.. [2] Kotz, Samuel, et. al. "The Laplace Distribution and
Generalizations, " Birkhauser, 2001.
.. [3] Weisstein, Eric W. "Laplace Distribution."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/LaplaceDistribution.html
.. [4] Wikipedia, "Laplace distribution",
http://en.wikipedia.org/wiki/Laplace_distribution
Examples
--------
Draw samples from the distribution
>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)
Plot Gaussian for comparison:
>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
... np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)
power(a, size=None)
Draws samples in [0, 1] from a power distribution with positive
exponent a - 1.
Also known as the power function distribution.
Parameters
----------
a : float or array_like of floats
Parameter of the distribution. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized power distribution.
Raises
------
ValueError
If a < 1.
Notes
-----
The probability density function is
.. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.
The power function distribution is just the inverse of the Pareto
distribution. It may also be seen as a special case of the Beta
distribution.
It is used, for example, in modeling the over-reporting of insurance
claims.
References
----------
.. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions
in economics and actuarial sciences", Wiley, 2003.
.. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:
Dataplot Reference Manual, Volume 2: Let Subcommands and Library
Functions", National Institute of Standards and Technology
Handbook Series, June 2003.
http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf
Examples
--------
Draw samples from the distribution:
>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()
Compare the power function distribution to the inverse of the Pareto.
>>> from scipy import stats
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)
>>> plt.figure()
>>> plt.hist(rvs, bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('np.random.power(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of 1 + np.random.pareto(5)')
>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
>>> plt.plot(xx,powpdf,'r-')
>>> plt.title('inverse of stats.pareto(5)')
weibull(a, size=None)
Draw samples from a Weibull distribution.
Draw samples from a 1-parameter Weibull distribution with the given
shape parameter `a`.
.. math:: X = (-ln(U))^{1/a}
Here, U is drawn from the uniform distribution over (0,1].
The more common 2-parameter Weibull, including a scale parameter
:math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.
Parameters
----------
a : float or array_like of floats
Shape of the distribution. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Weibull distribution.
See Also
--------
scipy.stats.weibull_max
scipy.stats.weibull_min
scipy.stats.genextreme
gumbel
Notes
-----
The Weibull (or Type III asymptotic extreme value distribution
for smallest values, SEV Type III, or Rosin-Rammler
distribution) is one of a class of Generalized Extreme Value
(GEV) distributions used in modeling extreme value problems.
This class includes the Gumbel and Frechet distributions.
The probability density for the Weibull distribution is
.. math:: p(x) = \frac{a}
{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
where :math:`a` is the shape and :math:`\lambda` the scale.
The function has its peak (the mode) at
:math:`\lambda(\frac{a-1}{a})^{1/a}`.
When ``a = 1``, the Weibull distribution reduces to the exponential
distribution.
References
----------
.. [1] Waloddi Weibull, Royal Technical University, Stockholm,
1939 "A Statistical Theory Of The Strength Of Materials",
Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,
Generalstabens Litografiska Anstalts Forlag, Stockholm.
.. [2] Waloddi Weibull, "A Statistical Distribution Function of
Wide Applicability", Journal Of Applied Mechanics ASME Paper
1951.
.. [3] Wikipedia, "Weibull distribution",
http://en.wikipedia.org/wiki/Weibull_distribution
Examples
--------
Draw samples from the distribution:
>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()
pareto(a, size=None)
Draw samples from a Pareto II or Lomax distribution with
specified shape.
The Lomax or Pareto II distribution is a shifted Pareto
distribution. The classical Pareto distribution can be
obtained from the Lomax distribution by adding 1 and
multiplying by the scale parameter ``m`` (see Notes). The
smallest value of the Lomax distribution is zero while for the
classical Pareto distribution it is ``mu``, where the standard
Pareto distribution has location ``mu = 1``. Lomax can also
be considered as a simplified version of the Generalized
Pareto distribution (available in SciPy), with the scale set
to one and the location set to zero.
The Pareto distribution must be greater than zero, and is
unbounded above. It is also known as the "80-20 rule". In
this distribution, 80 percent of the weights are in the lowest
20 percent of the range, while the other 20 percent fill the
remaining 80 percent of the range.
Parameters
----------
a : float or array_like of floats
Shape of the distribution. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` is a scalar. Otherwise,
``np.array(a).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Pareto distribution.
See Also
--------
scipy.stats.lomax : probability density function, distribution or
cumulative density function, etc.
scipy.stats.genpareto : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Pareto distribution is
.. math:: p(x) = \frac{am^a}{x^{a+1}}
where :math:`a` is the shape and :math:`m` the scale.
The Pareto distribution, named after the Italian economist
Vilfredo Pareto, is a power law probability distribution
useful in many real world problems. Outside the field of
economics it is generally referred to as the Bradford
distribution. Pareto developed the distribution to describe
the distribution of wealth in an economy. It has also found
use in insurance, web page access statistics, oil field sizes,
and many other problems, including the download frequency for
projects in Sourceforge [1]_. It is one of the so-called
"fat-tailed" distributions.
References
----------
.. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
Sourceforge projects.
.. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
.. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
Values, Birkhauser Verlag, Basel, pp 23-30.
.. [4] Wikipedia, "Pareto distribution",
http://en.wikipedia.org/wiki/Pareto_distribution
Examples
--------
Draw samples from the distribution:
>>> a, m = 3., 2. # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m
Display the histogram of the samples, along with the probability
density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, normed=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()
vonmises(mu, kappa, size=None)
Draw samples from a von Mises distribution.
Samples are drawn from a von Mises distribution with specified mode
(mu) and dispersion (kappa), on the interval [-pi, pi].
The von Mises distribution (also known as the circular normal
distribution) is a continuous probability distribution on the unit
circle. It may be thought of as the circular analogue of the normal
distribution.
Parameters
----------
mu : float or array_like of floats
Mode ("center") of the distribution.
kappa : float or array_like of floats
Dispersion of the distribution, has to be >=0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``mu`` and ``kappa`` are both scalars.
Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized von Mises distribution.
See Also
--------
scipy.stats.vonmises : probability density function, distribution, or
cumulative density function, etc.
Notes
-----
The probability density for the von Mises distribution is
.. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},
where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
and :math:`I_0(\kappa)` is the modified Bessel function of order 0.
The von Mises is named for Richard Edler von Mises, who was born in
Austria-Hungary, in what is now the Ukraine. He fled to the United
States in 1939 and became a professor at Harvard. He worked in
probability theory, aerodynamics, fluid mechanics, and philosophy of
science.
References
----------
.. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing," New York: Dover, 1972.
.. [2] von Mises, R., "Mathematical Theory of Probability
and Statistics", New York: Academic Press, 1964.
Examples
--------
Draw samples from the distribution:
>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0
>>> plt.hist(s, 50, normed=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))
>>> plt.plot(x, y, linewidth=2, color='r')
>>> plt.show()
standard_t(df, size=None)
Draw samples from a standard Student's t distribution with `df` degrees
of freedom.
A special case of the hyperbolic distribution. As `df` gets
large, the result resembles that of the standard normal
distribution (`standard_normal`).
Parameters
----------
df : int or array_like of ints
Degrees of freedom, should be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``df`` is a scalar. Otherwise,
``np.array(df).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized standard Student's t distribution.
Notes
-----
The probability density function for the t distribution is
.. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}
The t test is based on an assumption that the data come from a
Normal distribution. The t test provides a way to test whether
the sample mean (that is the mean calculated from the data) is
a good estimate of the true mean.
The derivation of the t-distribution was first published in
1908 by William Gosset while working for the Guinness Brewery
in Dublin. Due to proprietary issues, he had to publish under
a pseudonym, and so he used the name Student.
References
----------
.. [1] Dalgaard, Peter, "Introductory Statistics With R",
Springer, 2002.
.. [2] Wikipedia, "Student's t-distribution"
http://en.wikipedia.org/wiki/Student's_t-distribution
Examples
--------
From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
women in Kj is:
>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
... 7515, 8230, 8770])
Does their energy intake deviate systematically from the recommended
value of 7725 kJ?
We have 10 degrees of freedom, so is the sample mean within 95% of the
recommended value?
>>> s = np.random.standard_t(10, size=100000)
>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727
Calculate the t statistic, setting the ddof parameter to the unbiased
value so the divisor in the standard deviation will be degrees of
freedom, N-1.
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> import matplotlib.pyplot as plt
>>> h = plt.hist(s, bins=100, normed=True)
For a one-sided t-test, how far out in the distribution does the t
statistic appear?
>>> np.sum(s<t) / float(len(s))
0.0090699999999999999 #random
So the p-value is about 0.009, which says the null hypothesis has a
probability of about 99% of being true.
standard_cauchy(size=None)
Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
samples : ndarray or scalar
The drawn samples.
Notes
-----
The probability density function for the full Cauchy distribution is
.. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
(\frac{x-x_0}{\gamma})^2 \bigr] }
and the Standard Cauchy distribution just sets :math:`x_0=0` and
:math:`\gamma=1`
The Cauchy distribution arises in the solution to the driven harmonic
oscillator problem, and also describes spectral line broadening. It
also describes the distribution of values at which a line tilted at
a random angle will cut the x axis.
When studying hypothesis tests that assume normality, seeing how the
tests perform on data from a Cauchy distribution is a good indicator of
their sensitivity to a heavy-tailed distribution, since the Cauchy looks
very much like a Gaussian distribution, but with heavier tails.
References
----------
.. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy
Distribution",
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
.. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/CauchyDistribution.html
.. [3] Wikipedia, "Cauchy distribution"
http://en.wikipedia.org/wiki/Cauchy_distribution
Examples
--------
Draw samples and plot the distribution:
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()
noncentral_chisquare(df, nonc, size=None)
Draw samples from a noncentral chi-square distribution.
The noncentral :math:`\chi^2` distribution is a generalisation of
the :math:`\chi^2` distribution.
Parameters
----------
df : int or array_like of ints
Degrees of freedom, should be > 0 as of NumPy 1.10.0,
should be > 1 for earlier versions.
nonc : float or array_like of floats
Non-centrality, should be non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``df`` and ``nonc`` are both scalars.
Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized noncentral chi-square distribution.
Notes
-----
The probability density function for the noncentral Chi-square
distribution is
.. math:: P(x;df,nonc) = \sum^{\infty}_{i=0}
\frac{e^{-nonc/2}(nonc/2)^{i}}{i!}
\P_{Y_{df+2i}}(x),
where :math:`Y_{q}` is the Chi-square with q degrees of freedom.
In Delhi (2007), it is noted that the noncentral chi-square is
useful in bombing and coverage problems, the probability of
killing the point target given by the noncentral chi-squared
distribution.
References
----------
.. [1] Delhi, M.S. Holla, "On a noncentral chi-square distribution in
the analysis of weapon systems effectiveness", Metrika,
Volume 15, Number 1 / December, 1970.
.. [2] Wikipedia, "Noncentral chi-square distribution"
http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution
Examples
--------
Draw values from the distribution and plot the histogram
>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
... bins=200, normed=True)
>>> plt.show()
Draw values from a noncentral chisquare with very small noncentrality,
and compare to a chisquare.
>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
... bins=np.arange(0., 25, .1), normed=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
... bins=np.arange(0., 25, .1), normed=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()
Demonstrate how large values of non-centrality lead to a more symmetric
distribution.
>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
... bins=200, normed=True)
>>> plt.show()
chisquare(df, size=None)
Draw samples from a chi-square distribution.
When `df` independent random variables, each with standard normal
distributions (mean 0, variance 1), are squared and summed, the
resulting distribution is chi-square (see Notes). This distribution
is often used in hypothesis testing.
Parameters
----------
df : int or array_like of ints
Number of degrees of freedom.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``df`` is a scalar. Otherwise,
``np.array(df).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized chi-square distribution.
Raises
------
ValueError
When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
is given.
Notes
-----
The variable obtained by summing the squares of `df` independent,
standard normally distributed random variables:
.. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i
is chi-square distributed, denoted
.. math:: Q \sim \chi^2_k.
The probability density function of the chi-squared distribution is
.. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
x^{k/2 - 1} e^{-x/2},
where :math:`\Gamma` is the gamma function,
.. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.
References
----------
.. [1] NIST "Engineering Statistics Handbook"
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
Examples
--------
>>> np.random.chisquare(2,4)
array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272])
noncentral_f(dfnum, dfden, nonc, size=None)
Draw samples from the noncentral F distribution.
Samples are drawn from an F distribution with specified parameters,
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
freedom in denominator), where both parameters > 1.
`nonc` is the non-centrality parameter.
Parameters
----------
dfnum : int or array_like of ints
Parameter, should be > 1.
dfden : int or array_like of ints
Parameter, should be > 1.
nonc : float or array_like of floats
Parameter, should be >= 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``dfnum``, ``dfden``, and ``nonc``
are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size``
samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized noncentral Fisher distribution.
Notes
-----
When calculating the power of an experiment (power = probability of
rejecting the null hypothesis when a specific alternative is true) the
non-central F statistic becomes important. When the null hypothesis is
true, the F statistic follows a central F distribution. When the null
hypothesis is not true, then it follows a non-central F statistic.
References
----------
.. [1] Weisstein, Eric W. "Noncentral F-Distribution."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/NoncentralF-Distribution.html
.. [2] Wikipedia, "Noncentral F-distribution",
http://en.wikipedia.org/wiki/Noncentral_F-distribution
Examples
--------
In a study, testing for a specific alternative to the null hypothesis
requires use of the Noncentral F distribution. We need to calculate the
area in the tail of the distribution that exceeds the value of the F
distribution for the null hypothesis. We'll plot the two probability
distributions for comparison.
>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, normed=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, normed=True)
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()
f(dfnum, dfden, size=None)
Draw samples from an F distribution.
Samples are drawn from an F distribution with specified parameters,
`dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
freedom in denominator), where both parameters should be greater than
zero.
The random variate of the F distribution (also known as the
Fisher distribution) is a continuous probability distribution
that arises in ANOVA tests, and is the ratio of two chi-square
variates.
Parameters
----------
dfnum : int or array_like of ints
Degrees of freedom in numerator. Should be greater than zero.
dfden : int or array_like of ints
Degrees of freedom in denominator. Should be greater than zero.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``dfnum`` and ``dfden`` are both scalars.
Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized Fisher distribution.
See Also
--------
scipy.stats.f : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The F statistic is used to compare in-group variances to between-group
variances. Calculating the distribution depends on the sampling, and
so it is a function of the respective degrees of freedom in the
problem. The variable `dfnum` is the number of samples minus one, the
between-groups degrees of freedom, while `dfden` is the within-groups
degrees of freedom, the sum of the number of samples in each group
minus the number of groups.
References
----------
.. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
Fifth Edition, 2002.
.. [2] Wikipedia, "F-distribution",
http://en.wikipedia.org/wiki/F-distribution
Examples
--------
An example from Glantz[1], pp 47-40:
Two groups, children of diabetics (25 people) and children from people
without diabetes (25 controls). Fasting blood glucose was measured,
case group had a mean value of 86.1, controls had a mean value of
82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
data consistent with the null hypothesis that the parents diabetic
status does not affect their children's blood glucose levels?
Calculating the F statistic from the data gives a value of 36.01.
Draw samples from the distribution:
>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)
The lower bound for the top 1% of the samples is :
>>> sort(s)[-10]
7.61988120985
So there is about a 1% chance that the F statistic will exceed 7.62,
the measured value is 36, so the null hypothesis is rejected at the 1%
level.
gamma(shape, scale=1.0, size=None)
Draw samples from a Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters,
`shape` (sometimes designated "k") and `scale` (sometimes designated
"theta"), where both parameters are > 0.
Parameters
----------
shape : float or array_like of floats
The shape of the gamma distribution. Should be greater than zero.
scale : float or array_like of floats, optional
The scale of the gamma distribution. Should be greater than zero.
Default is equal to 1.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``shape`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized gamma distribution.
See Also
--------
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gamma distribution is
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
where :math:`k` is the shape and :math:`\theta` the scale,
and :math:`\Gamma` is the Gamma function.
The Gamma distribution is often used to model the times to failure of
electronic components, and arises naturally in processes for which the
waiting times between Poisson distributed events are relevant.
References
----------
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/GammaDistribution.html
.. [2] Wikipedia, "Gamma distribution",
http://en.wikipedia.org/wiki/Gamma_distribution
Examples
--------
Draw samples from the distribution:
>>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps
>>> count, bins, ignored = plt.hist(s, 50, normed=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /
... (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()
standard_gamma(shape, size=None)
Draw samples from a standard Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters,
shape (sometimes designated "k") and scale=1.
Parameters
----------
shape : float or array_like of floats
Parameter, should be > 0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``shape`` is a scalar. Otherwise,
``np.array(shape).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized standard gamma distribution.
See Also
--------
scipy.stats.gamma : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gamma distribution is
.. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},
where :math:`k` is the shape and :math:`\theta` the scale,
and :math:`\Gamma` is the Gamma function.
The Gamma distribution is often used to model the times to failure of
electronic components, and arises naturally in processes for which the
waiting times between Poisson distributed events are relevant.
References
----------
.. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/GammaDistribution.html
.. [2] Wikipedia, "Gamma distribution",
http://en.wikipedia.org/wiki/Gamma_distribution
Examples
--------
Draw samples from the distribution:
>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps
>>> count, bins, ignored = plt.hist(s, 50, normed=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ \
... (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')
>>> plt.show()
standard_exponential(size=None)
Draw samples from the standard exponential distribution.
`standard_exponential` is identical to the exponential distribution
with a scale parameter of 1.
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : float or ndarray
Drawn samples.
Examples
--------
Output a 3x8000 array:
>>> n = np.random.standard_exponential((3, 8000))
exponential(scale=1.0, size=None)
Draw samples from an exponential distribution.
Its probability density function is
.. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),
for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter,
which is the inverse of the rate parameter :math:`\lambda = 1/\beta`.
The rate parameter is an alternative, widely used parameterization
of the exponential distribution [3]_.
The exponential distribution is a continuous analogue of the
geometric distribution. It describes many common situations, such as
the size of raindrops measured over many rainstorms [1]_, or the time
between page requests to Wikipedia [2]_.
Parameters
----------
scale : float or array_like of floats
The scale parameter, :math:`\beta = 1/\lambda`.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``scale`` is a scalar. Otherwise,
``np.array(scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized exponential distribution.
References
----------
.. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and
Random Signal Principles", 4th ed, 2001, p. 57.
.. [2] Wikipedia, "Poisson process",
http://en.wikipedia.org/wiki/Poisson_process
.. [3] Wikipedia, "Exponential distribution",
http://en.wikipedia.org/wiki/Exponential_distribution
beta(a, b, size=None)
Draw samples from a Beta distribution.
The Beta distribution is a special case of the Dirichlet distribution,
and is related to the Gamma distribution. It has the probability
distribution function
.. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1}
(1 - x)^{\beta - 1},
where the normalisation, B, is the beta function,
.. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1}
(1 - t)^{\beta - 1} dt.
It is often seen in Bayesian inference and order statistics.
Parameters
----------
a : float or array_like of floats
Alpha, non-negative.
b : float or array_like of floats
Beta, non-negative.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``a`` and ``b`` are both scalars.
Otherwise, ``np.broadcast(a, b).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized beta distribution.
normal(loc=0.0, scale=1.0, size=None)
Draw random samples from a normal (Gaussian) distribution.
The probability density function of the normal distribution, first
derived by De Moivre and 200 years later by both Gauss and Laplace
independently [2]_, is often called the bell curve because of
its characteristic shape (see the example below).
The normal distributions occurs often in nature. For example, it
describes the commonly occurring distribution of samples influenced
by a large number of tiny, random disturbances, each with its own
unique distribution [2]_.
Parameters
----------
loc : float or array_like of floats
Mean ("centre") of the distribution.
scale : float or array_like of floats
Standard deviation (spread or "width") of the distribution.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized normal distribution.
See Also
--------
scipy.stats.norm : probability density function, distribution or
cumulative density function, etc.
Notes
-----
The probability density for the Gaussian distribution is
.. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },
where :math:`\mu` is the mean and :math:`\sigma` the standard
deviation. The square of the standard deviation, :math:`\sigma^2`,
is called the variance.
The function has its peak at the mean, and its "spread" increases with
the standard deviation (the function reaches 0.607 times its maximum at
:math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that
`numpy.random.normal` is more likely to return samples lying close to
the mean, rather than those far away.
References
----------
.. [1] Wikipedia, "Normal distribution",
http://en.wikipedia.org/wiki/Normal_distribution
.. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
Random Variables and Random Signal Principles", 4th ed., 2001,
pp. 51, 51, 125.
Examples
--------
Draw samples from the distribution:
>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)
Verify the mean and the variance:
>>> abs(mu - np.mean(s)) < 0.01
True
>>> abs(sigma - np.std(s, ddof=1)) < 0.01
True
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
... np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
... linewidth=2, color='r')
>>> plt.show()
standard_normal(size=None)
Draw samples from a standard Normal distribution (mean=0, stdev=1).
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : float or ndarray
Drawn samples.
Examples
--------
>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, #random
-0.38672696, -0.4685006 ]) #random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)
random_integers(low, high=None, size=None)
Random integers of type np.int between `low` and `high`, inclusive.
Return random integers of type np.int from the "discrete uniform"
distribution in the closed interval [`low`, `high`]. If `high` is
None (the default), then results are from [1, `low`]. The np.int
type translates to the C long type used by Python 2 for "short"
integers and its precision is platform dependent.
This function has been deprecated. Use randint instead.
.. deprecated:: 1.11.0
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution (unless
``high=None``, in which case this parameter is the *highest* such
integer).
high : int, optional
If provided, the largest (signed) integer to be drawn from the
distribution (see above for behavior if ``high=None``).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : int or ndarray of ints
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
See Also
--------
random.randint : Similar to `random_integers`, only for the half-open
interval [`low`, `high`), and 0 is the lowest value if `high` is
omitted.
Notes
-----
To sample from N evenly spaced floating-point numbers between a and b,
use::
a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)
Examples
--------
>>> np.random.random_integers(5)
4
>>> type(np.random.random_integers(5))
<type 'int'>
>>> np.random.random_integers(5, size=(3.,2.))
array([[5, 4],
[3, 3],
[4, 5]])
Choose five random numbers from the set of five evenly-spaced
numbers between 0 and 2.5, inclusive (*i.e.*, from the set
:math:`{0, 5/8, 10/8, 15/8, 20/8}`):
>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625, 1.25 , 0.625, 0.625, 2.5 ])
Roll two six sided dice 1000 times and sum the results:
>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2
Display results as a histogram:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, normed=True)
>>> plt.show()
randn(d0, d1, ..., dn)
Return a sample (or samples) from the "standard normal" distribution.
If positive, int_like or int-convertible arguments are provided,
`randn` generates an array of shape ``(d0, d1, ..., dn)``, filled
with random floats sampled from a univariate "normal" (Gaussian)
distribution of mean 0 and variance 1 (if any of the :math:`d_i` are
floats, they are first converted to integers by truncation). A single
float randomly sampled from the distribution is returned if no
argument is provided.
This is a convenience function. If you want an interface that takes a
tuple as the first argument, use `numpy.random.standard_normal` instead.
Parameters
----------
d0, d1, ..., dn : int, optional
The dimensions of the returned array, should be all positive.
If no argument is given a single Python float is returned.
Returns
-------
Z : ndarray or float
A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from
the standard normal distribution, or a single such float if
no parameters were supplied.
See Also
--------
random.standard_normal : Similar, but takes a tuple as its argument.
Notes
-----
For random samples from :math:`N(\mu, \sigma^2)`, use:
``sigma * np.random.randn(...) + mu``
Examples
--------
>>> np.random.randn()
2.1923875335537315 #random
Two-by-four array of samples from N(3, 6.25):
>>> 2.5 * np.random.randn(2, 4) + 3
array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], #random
[ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) #random
rand(d0, d1, ..., dn)
Random values in a given shape.
Create an array of the given shape and populate it with
random samples from a uniform distribution
over ``[0, 1)``.
Parameters
----------
d0, d1, ..., dn : int, optional
The dimensions of the returned array, should all be positive.
If no argument is given a single Python float is returned.
Returns
-------
out : ndarray, shape ``(d0, d1, ..., dn)``
Random values.
See Also
--------
random
Notes
-----
This is a convenience function. If you want an interface that
takes a shape-tuple as the first argument, refer to
np.random.random_sample .
Examples
--------
>>> np.random.rand(3,2)
array([[ 0.14022471, 0.96360618], #random
[ 0.37601032, 0.25528411], #random
[ 0.49313049, 0.94909878]]) #random
uniform(low=0.0, high=1.0, size=None)
Draw samples from a uniform distribution.
Samples are uniformly distributed over the half-open interval
``[low, high)`` (includes low, but excludes high). In other words,
any value within the given interval is equally likely to be drawn
by `uniform`.
Parameters
----------
low : float or array_like of floats, optional
Lower boundary of the output interval. All values generated will be
greater than or equal to low. The default value is 0.
high : float or array_like of floats
Upper boundary of the output interval. All values generated will be
less than high. The default value is 1.0.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. If size is ``None`` (default),
a single value is returned if ``low`` and ``high`` are both scalars.
Otherwise, ``np.broadcast(low, high).size`` samples are drawn.
Returns
-------
out : ndarray or scalar
Drawn samples from the parameterized uniform distribution.
See Also
--------
randint : Discrete uniform distribution, yielding integers.
random_integers : Discrete uniform distribution over the closed
interval ``[low, high]``.
random_sample : Floats uniformly distributed over ``[0, 1)``.
random : Alias for `random_sample`.
rand : Convenience function that accepts dimensions as input, e.g.,
``rand(2,2)`` would generate a 2-by-2 array of floats,
uniformly distributed over ``[0, 1)``.
Notes
-----
The probability density function of the uniform distribution is
.. math:: p(x) = \frac{1}{b - a}
anywhere within the interval ``[a, b)``, and zero elsewhere.
When ``high`` == ``low``, values of ``low`` will be returned.
If ``high`` < ``low``, the results are officially undefined
and may eventually raise an error, i.e. do not rely on this
function to behave when passed arguments satisfying that
inequality condition.
Examples
--------
Draw samples from the distribution:
>>> s = np.random.uniform(-1,0,1000)
All values are within the given interval:
>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True
Display the histogram of the samples, along with the
probability density function:
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, normed=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()
choice(a, size=None, replace=True, p=None)
Generates a random sample from a given 1-D array
.. versionadded:: 1.7.0
Parameters
-----------
a : 1-D array-like or int
If an ndarray, a random sample is generated from its elements.
If an int, the random sample is generated as if a were np.arange(a)
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
replace : boolean, optional
Whether the sample is with or without replacement
p : 1-D array-like, optional
The probabilities associated with each entry in a.
If not given the sample assumes a uniform distribution over all
entries in a.
Returns
--------
samples : single item or ndarray
The generated random samples
Raises
-------
ValueError
If a is an int and less than zero, if a or p are not 1-dimensional,
if a is an array-like of size 0, if p is not a vector of
probabilities, if a and p have different lengths, or if
replace=False and the sample size is greater than the population
size
See Also
---------
randint, shuffle, permutation
Examples
---------
Generate a uniform random sample from np.arange(5) of size 3:
>>> np.random.choice(5, 3)
array([0, 3, 4])
>>> #This is equivalent to np.random.randint(0,5,3)
Generate a non-uniform random sample from np.arange(5) of size 3:
>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0])
Generate a uniform random sample from np.arange(5) of size 3 without
replacement:
>>> np.random.choice(5, 3, replace=False)
array([3,1,0])
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]
Generate a non-uniform random sample from np.arange(5) of size
3 without replacement:
>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0])
Any of the above can be repeated with an arbitrary array-like
instead of just integers. For instance:
>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'],
dtype='|S11')
bytes(length)
Return random bytes.
Parameters
----------
length : int
Number of random bytes.
Returns
-------
out : str
String of length `length`.
Examples
--------
>>> np.random.bytes(10)
' eh\x85\x022SZ\xbf\xa4' #random
randint(low, high=None, size=None, dtype='l')
Return random integers from `low` (inclusive) to `high` (exclusive).
Return random integers from the "discrete uniform" distribution of
the specified dtype in the "half-open" interval [`low`, `high`). If
`high` is None (the default), then results are from [0, `low`).
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution (unless
``high=None``, in which case this parameter is one above the
*highest* such integer).
high : int, optional
If provided, one above the largest (signed) integer to be drawn
from the distribution (see above for behavior if ``high=None``).
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
dtype : dtype, optional
Desired dtype of the result. All dtypes are determined by their
name, i.e., 'int64', 'int', etc, so byteorder is not available
and a specific precision may have different C types depending
on the platform. The default value is 'np.int'.
.. versionadded:: 1.11.0
Returns
-------
out : int or ndarray of ints
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
See Also
--------
random.random_integers : similar to `randint`, only for the closed
interval [`low`, `high`], and 1 is the lowest value if `high` is
omitted. In particular, this other one is the one to use to generate
uniformly distributed discrete non-integers.
Examples
--------
>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
Generate a 2 x 4 array of ints between 0 and 4, inclusive:
>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1],
[3, 2, 2, 0]])
tomaxint(size=None)
Random integers between 0 and ``sys.maxint``, inclusive.
Return a sample of uniformly distributed random integers in the interval
[0, ``sys.maxint``].
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : ndarray
Drawn samples, with shape `size`.
See Also
--------
randint : Uniform sampling over a given half-open interval of integers.
random_integers : Uniform sampling over a given closed interval of
integers.
Examples
--------
>>> RS = np.random.mtrand.RandomState() # need a RandomState object
>>> RS.tomaxint((2,2,2))
array([[[1170048599, 1600360186],
[ 739731006, 1947757578]],
[[1871712945, 752307660],
[1601631370, 1479324245]]])
>>> import sys
>>> sys.maxint
2147483647
>>> RS.tomaxint((2,2,2)) < sys.maxint
array([[[ True, True],
[ True, True]],
[[ True, True],
[ True, True]]], dtype=bool)
random_sample(size=None)
Return random floats in the half-open interval [0.0, 1.0).
Results are from the "continuous uniform" distribution over the
stated interval. To sample :math:`Unif[a, b), b > a` multiply
the output of `random_sample` by `(b-a)` and add `a`::
(b - a) * random_sample() + a
Parameters
----------
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
Returns
-------
out : float or ndarray of floats
Array of random floats of shape `size` (unless ``size=None``, in which
case a single float is returned).
Examples
--------
>>> np.random.random_sample()
0.47108547995356098
>>> type(np.random.random_sample())
<type 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428])
Three-by-two array of random numbers from [-5, 0):
>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984],
[-2.99091858, -0.79479508],
[-1.23204345, -1.75224494]])
set_state(state)
Set the internal state of the generator from a tuple.
For use if one has reason to manually (re-)set the internal state of the
"Mersenne Twister"[1]_ pseudo-random number generating algorithm.
Parameters
----------
state : tuple(str, ndarray of 624 uints, int, int, float)
The `state` tuple has the following items:
1. the string 'MT19937', specifying the Mersenne Twister algorithm.
2. a 1-D array of 624 unsigned integers ``keys``.
3. an integer ``pos``.
4. an integer ``has_gauss``.
5. a float ``cached_gaussian``.
Returns
-------
out : None
Returns 'None' on success.
See Also
--------
get_state
Notes
-----
`set_state` and `get_state` are not needed to work with any of the
random distributions in NumPy. If the internal state is manually altered,
the user should know exactly what he/she is doing.
For backwards compatibility, the form (str, array of 624 uints, int) is
also accepted although it is missing some information about the cached
Gaussian value: ``state = ('MT19937', keys, pos)``.
References
----------
.. [1] M. Matsumoto and T. Nishimura, "Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number
generator," *ACM Trans. on Modeling and Computer Simulation*,
Vol. 8, No. 1, pp. 3-30, Jan. 1998.
get_state()
Return a tuple representing the internal state of the generator.
For more details, see `set_state`.
Returns
-------
out : tuple(str, ndarray of 624 uints, int, int, float)
The returned tuple has the following items:
1. the string 'MT19937'.
2. a 1-D array of 624 unsigned integer keys.
3. an integer ``pos``.
4. an integer ``has_gauss``.
5. a float ``cached_gaussian``.
See Also
--------
set_state
Notes
-----
`set_state` and `get_state` are not needed to work with any of the
random distributions in NumPy. If the internal state is manually altered,
the user should know exactly what he/she is doing.
seed(seed=None)
Seed the generator.
This method is called when `RandomState` is initialized. It can be
called again to re-seed the generator. For details, see `RandomState`.
Parameters
----------
seed : int or array_like, optional
Seed for `RandomState`.
Must be convertible to 32 bit unsigned integers.
See Also
--------
RandomState
_rand_uint64(low, high, size, rngstate)
Return random np.uint64 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.uint64 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.uint64
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_uint32(low, high, size, rngstate)
Return random np.uint32 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.uint32 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.uint32
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_uint16(low, high, size, rngstate)
Return random np.uint16 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.uint16 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.uint16
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_uint8(low, high, size, rngstate)
Return random np.uint8 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.uint8 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.uint8
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_int64(low, high, size, rngstate)
Return random np.int64 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.int64 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.int64
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_int32(low, high, size, rngstate)
Return random np.int32 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.int32 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.int32
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_int16(low, high, size, rngstate)
Return random np.int16 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.int16 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.int16
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_int8(low, high, size, rngstate)
Return random np.int8 integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.int8 type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.int8
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
_rand_bool(low, high, size, rngstate)
Return random np.bool_ integers between ``low`` and ``high``, inclusive.
Return random integers from the "discrete uniform" distribution in the
closed interval [``low``, ``high``). On entry the arguments are presumed
to have been validated for size and order for the np.bool_ type.
Parameters
----------
low : int
Lowest (signed) integer to be drawn from the distribution.
high : int
Highest (signed) integer to be drawn from the distribution.
size : int or tuple of ints
Output shape. If the given shape is, e.g., ``(m, n, k)``, then
``m * n * k`` samples are drawn. Default is None, in which case a
single value is returned.
rngstate : encapsulated pointer to rk_state
The specific type depends on the python version. In Python 2 it is
a PyCObject, in Python 3 a PyCapsule object.
Returns
-------
out : python integer or ndarray of np.bool_
`size`-shaped array of random integers from the appropriate
distribution, or a single such random int if `size` not provided.
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