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# scipy.stats.poisson_binom

scipy.stats.poisson_binom = <scipy.stats._discrete_distns.poisson_binom_gen object>

  A Poisson Binomial discrete random variable.

  As an instance of the `rv_discrete` class, `poisson_binom` 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.

  Methods

  rvs(p, loc=0, size=1, random_state=None) | Random variates.
  pmf(k, p, loc=0) | Probability mass function.
  logpmf(k, p, loc=0) | Log of the probability mass function.
  cdf(k, p, loc=0) | Cumulative distribution function.
  logcdf(k, p, loc=0) | Log of the cumulative distribution function.
  sf(k, p, loc=0) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(k, p, loc=0) | Log of the survival function.
  ppf(q, p, loc=0) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, p, loc=0) | Inverse survival function (inverse of `sf`).
  stats(p, loc=0, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(p, loc=0) | (Differential) entropy of the RV.
  expect(func, args=(p,), loc=0, lb=None, ub=None, conditional=False) | Expected value of a function (of one argument) with respect to the distribution.
  median(p, loc=0) | Median of the distribution.
  mean(p, loc=0) | Mean of the distribution.
  var(p, loc=0) | Variance of the distribution.
  std(p, loc=0) | Standard deviation of the distribution.
  interval(confidence, p, loc=0) | Confidence interval with equal areas around the median.

  See also

  `binom`

  Notes

  The probability mass function for `poisson_binom` is:

  \[f(k; p_1, p_2, ..., p_n) = \sum_{A \in F_k} \prod_{i \in A} p_i \prod_{j \in A^C} 1 - p_j\]

  where \(k \in \{0, 1, \dots, n-1, n\}\), \(F_k\) is the set of all subsets of \(k\) integers that can be selected \(\{0, 1, \dots, n-1, n\}\), and \(A^C\) is the complement of a set \(A\).

  `poisson_binom` accepts a single array argument `p` for shape parameters \(0 ≤ p_i ≤ 1\), where the last axis corresponds with the index \(i\) and any others are for batch dimensions. Broadcasting behaves according to the usual rules except that the last axis of `p` is ignored. Instances of this class do not support serialization/unserialization.

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

  References

  [1]

  “Poisson binomial distribution”, Wikipedia, https://en.wikipedia.org/wiki/Poisson_binomial_distribution

  [2]

  Biscarri, William, Sihai Dave Zhao, and Robert J. Brunner. “A simple and fast method for computing the Poisson binomial distribution function”. Computational Statistics & Data Analysis 122 (2018) 92-100. DOI:10.1016/j.csda.2018.01.007

  Examples

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

  Get the support:

      >>> p = [0.1, 0.6, 0.7, 0.8]
      >>> lb, ub = poisson_binom.support(p)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = poisson_binom.stats(p, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(poisson_binom.ppf(0.01, p),
      ...               poisson_binom.ppf(0.99, p))
      >>> ax.plot(x, poisson_binom.pmf(x, p), 'bo', ms=8, label='poisson_binom pmf')
      >>> ax.vlines(x, 0, poisson_binom.pmf(x, p), 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_binom(p)
      >>> 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_binom.cdf(x, p)
      >>> np.allclose(x, poisson_binom.ppf(prob, p))
      True

  Generate random numbers:

      >>> r = poisson_binom.rvs(p, size=1000)