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# scipy.stats.zipf

scipy.stats.zipf = <scipy.stats._discrete_distns.zipf_gen object>

  A Zipf (Zeta) discrete random variable.

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

  See also

  `zipfian`

  Notes

  The probability mass function for `zipf` is:

  \[f(k, a) = \frac{1}{\zeta(a) k^a}\]

  for \(k \ge 1\), \(a > 1\).

  `zipf` takes \(a > 1\) as shape parameter. \(\zeta\) is the Riemann zeta function (`scipy.special.zeta`)

  The Zipf distribution is also known as the zeta distribution, which is a special case of the Zipfian distribution (`zipfian`).

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

  References

  [1]

  “Zeta Distribution”, Wikipedia, https://en.wikipedia.org/wiki/Zeta_distribution

  Examples

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

  Get the support:

      >>> a = 6.6
      >>> lb, ub = zipf.support(a)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = zipf.stats(a, moments='mvsk')

  Display the probability mass function (`pmf`):

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

  Generate random numbers:

      >>> r = zipf.rvs(a, size=1000)

  Confirm that `zipf` is the large n limit of `zipfian`.

      >>> import numpy as np
      >>> from scipy.stats import zipf, zipfian
      >>> k = np.arange(11)
      >>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000))
      True