scipy.stats.poisson = <scipy.stats._discrete_distns.poisson_gen object>[source]#

A Poisson discrete random variable.

As an instance of the rv_discrete class, poisson object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.


The probability mass function for poisson is:

\[f(k) = \exp(-\mu) \frac{\mu^k}{k!}\]

for \(k \ge 0\).

poisson takes \(\mu \geq 0\) as shape parameter. When \(\mu = 0\), the pmf method returns 1.0 at quantile \(k = 0\).

The probability mass function above is defined in the “standardized” form. To shift distribution use the loc parameter. Specifically, poisson.pmf(k, mu, loc) is identically equivalent to poisson.pmf(k - loc, mu).


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

Calculate the first four 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, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed.

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)

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_state=None)

Random variates.

pmf(k, mu, loc=0)

Probability mass function.

logpmf(k, mu, loc=0)

Log of the probability mass function.

cdf(k, mu, loc=0)

Cumulative distribution function.

logcdf(k, mu, loc=0)

Log of the cumulative distribution function.

sf(k, mu, loc=0)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(k, 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, args=(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(confidence, mu, loc=0)

Confidence interval with equal areas around the median.