scipy.stats.poisson = <scipy.stats.distributions.poisson_gen object at 0x4ad9b50>[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:

Parameters :

x : array_like


q : array_like

lower or upper tail probability

mu : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

scale : array_like, optional

scale parameter (default=1)

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
>>> [ mu ] = [<Replace with reasonable values>]
>>> rv = poisson(mu)

Display frozen pmf

>>> x = np.arange(0, np.minimum(rv.dist.b, 3))
>>> h = plt.vlines(x, 0, rv.pmf(x), lw=2)

Here, rv.dist.b is the right endpoint of the support of rv.dist.

Check accuracy of cdf and ppf

>>> prb = poisson.cdf(x, mu)
>>> h = plt.semilogy(np.abs(x - poisson.ppf(prb, mu)) + 1e-20)

Random number generation

>>> R = poisson.rvs(mu, size=100)


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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