Skip to main content
`Ctrl`+`K`

# scipy.stats.skellam

scipy.stats.skellam = <scipy.stats._discrete_distns.skellam_gen object>

  A Skellam discrete random variable.

  As an instance of the `rv_discrete` class, `skellam` 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(mu1, mu2, loc=0, size=1, random_state=None) | Random variates.
  pmf(k, mu1, mu2, loc=0) | Probability mass function.
  logpmf(k, mu1, mu2, loc=0) | Log of the probability mass function.
  cdf(k, mu1, mu2, loc=0) | Cumulative distribution function.
  logcdf(k, mu1, mu2, loc=0) | Log of the cumulative distribution function.
  sf(k, mu1, mu2, loc=0) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(k, mu1, mu2, loc=0) | Log of the survival function.
  ppf(q, mu1, mu2, loc=0) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, mu1, mu2, loc=0) | Inverse survival function (inverse of `sf`).
  stats(mu1, mu2, loc=0, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(mu1, mu2, loc=0) | (Differential) entropy of the RV.
  expect(func, args=(mu1, mu2), loc=0, lb=None, ub=None, conditional=False) | Expected value of a function (of one argument) with respect to the distribution.
  median(mu1, mu2, loc=0) | Median of the distribution.
  mean(mu1, mu2, loc=0) | Mean of the distribution.
  var(mu1, mu2, loc=0) | Variance of the distribution.
  std(mu1, mu2, loc=0) | Standard deviation of the distribution.
  interval(confidence, mu1, mu2, loc=0) | Confidence interval with equal areas around the median.

  Notes

  Probability distribution of the difference of two correlated or uncorrelated Poisson random variables.

  Let \(k_1\) and \(k_2\) be two Poisson-distributed r.v. with expected values \(\lambda_1\) and \(\lambda_2\). Then, \(k_1 - k_2\) follows a Skellam distribution with parameters \(\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}\) and \(\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}\), where \(\rho\) is the correlation coefficient between \(k_1\) and \(k_2\). If the two Poisson-distributed r.v. are independent then \(\rho = 0\).

  Parameters \(\mu_1\) and \(\mu_2\) must be strictly positive.

  `skellam` takes \(\mu_1\) and \(\mu_2\) as shape parameters.

  The probability mass function above is defined in the “standardized” form. To shift distribution use the `loc` parameter. Specifically, `skellam.pmf(k, mu1, mu2, loc)` is identically equivalent to `skellam.pmf(k - loc, mu1, mu2)`.

  References

  [1]

  Skellam, J. G. “The Frequency Distribution of the Difference Between Two Poisson Variates Belonging to Different Populations.” Journal of the Royal Statistical Society 109, no. 3 (1946): 296-296. DOI:10.2307/2981372

  [2]

  “Skellam distribution”, Wikipedia, https://en.wikipedia.org/wiki/Skellam_distribution

  Examples

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

  Get the support:

      >>> mu1, mu2 = 15, 8
      >>> lb, ub = skellam.support(mu1, mu2)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = skellam.stats(mu1, mu2, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(skellam.ppf(0.01, mu1, mu2),
      ...               skellam.ppf(0.99, mu1, mu2))
      >>> ax.plot(x, skellam.pmf(x, mu1, mu2), 'bo', ms=8, label='skellam pmf')
      >>> ax.vlines(x, 0, skellam.pmf(x, mu1, mu2), 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 = skellam(mu1, mu2)
      >>> 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 = skellam.cdf(x, mu1, mu2)
      >>> np.allclose(x, skellam.ppf(prob, mu1, mu2))
      True

  Generate random numbers:

      >>> r = skellam.rvs(mu1, mu2, size=1000)