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# scipy.stats.poisson

scipy.stats.poisson = <scipy.stats._discrete_distns.poisson_gen object>

  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.

  Notes

  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)`.

  Examples

      >>> 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)
      >>> plt.show()

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

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

  Methods

  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.