scipy.stats.poisson = <scipy.stats._discrete_distns.poisson_gen object at 0x2b45d2fe2550>[source]

A Poisson discrete random variable.

Discrete random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:


x : array_like


q : array_like

lower or upper tail probability

mu : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : str, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)

Alternatively, the object may be called (as a function) to fix the shape and

location parameters returning a “frozen” discrete RV object:

rv = poisson(mu, loc=0)

  • Frozen RV object with the same methods but holding the given shape and location fixed.


The probability mass function for poisson is:

poisson.pmf(k) = exp(-mu) * mu**k / k!

for k >= 0.

poisson takes mu as shape parameter.


>>> from scipy.stats import poisson
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate a few first moments:

>>> mu = 0.6
>>> mean, var, skew, kurt = poisson.stats(mu, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(poisson.ppf(0.01, mu),
...               poisson.ppf(0.99, mu))
>>> ax.plot(x, poisson.pmf(x, mu), 'bo', ms=8, label='poisson pmf')
>>> ax.vlines(x, 0, poisson.pmf(x, mu), colors='b', lw=5, alpha=0.5)

Alternatively, freeze the distribution and display the frozen pmf:

>>> rv = poisson(mu)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
...         label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)

(Source code)


Check accuracy of cdf and ppf:

>>> prob = poisson.cdf(x, mu)
>>> np.allclose(x, poisson.ppf(prob, mu))

Generate random numbers:

>>> r = poisson.rvs(mu, size=1000)


rvs(mu, loc=0, size=1) Random variates.
pmf(x, mu, loc=0) Probability mass function.
logpmf(x, mu, loc=0) Log of the probability mass function.
cdf(x, mu, loc=0) Cumulative density function.
logcdf(x, mu, loc=0) Log of the cumulative density function.
sf(x, mu, loc=0) Survival function (1-cdf — sometimes more accurate).
logsf(x, mu, loc=0) Log of the survival function.
ppf(q, mu, loc=0) Percent point function (inverse of cdf — percentiles).
isf(q, mu, loc=0) Inverse survival function (inverse of sf).
stats(mu, loc=0, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(mu, loc=0) (Differential) entropy of the RV.
expect(func, mu, loc=0, lb=None, ub=None, conditional=False) Expected value of a function (of one argument) with respect to the distribution.
median(mu, loc=0) Median of the distribution.
mean(mu, loc=0) Mean of the distribution.
var(mu, loc=0) Variance of the distribution.
std(mu, loc=0) Standard deviation of the distribution.
interval(alpha, mu, loc=0) Endpoints of the range that contains alpha percent of the distribution

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