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# scipy.stats.powerlaw

scipy.stats.powerlaw = <scipy.stats._continuous_distns.powerlaw_gen object>

  A power-function continuous random variable.

  As an instance of the `rv_continuous` class, `powerlaw` 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, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, a, loc=0, scale=1) | Probability density function.
  logpdf(x, a, loc=0, scale=1) | Log of the probability density function.
  cdf(x, a, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, a, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, a, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, a, loc=0, scale=1) | Log of the survival function.
  ppf(q, a, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, a, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, a, loc=0, scale=1) | Non-central moment of the specified order.
  stats(a, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(a, loc=0, scale=1) | (Differential) entropy of the RV.
  fit(data) | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
  expect(func, args=(a,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) | Expected value of a function (of one argument) with respect to the distribution.
  median(a, loc=0, scale=1) | Median of the distribution.
  mean(a, loc=0, scale=1) | Mean of the distribution.
  var(a, loc=0, scale=1) | Variance of the distribution.
  std(a, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, a, loc=0, scale=1) | Confidence interval with equal areas around the median.

  See also

  `pareto`

  Notes

  The probability density function for `powerlaw` is:

  \[f(x, a) = a x^{a-1}\]

  for \(0 \le x \le 1\), \(a > 0\).

  `powerlaw` takes `a` as a shape parameter for \(a\).

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `powerlaw.pdf(x, a, loc, scale)` is identically equivalent to `powerlaw.pdf(y, a) / scale` with `y = (x - loc) / scale`. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

  For example, the support of `powerlaw` can be adjusted from the default interval `[0, 1]` to the interval `[c, c+d]` by setting `loc=c` and `scale=d`. For a power-law distribution with infinite support, see `pareto`. For a power-law distribution described by PDF:

  \[f(x; a, l, h) = \frac{a}{h^a - l^2} x^{a-1}\]

  with \(a \neq 0\) and \(0 < l < x < h\), see `truncpareto`.

  `powerlaw` is a special case of `beta` with `b=1`.

  Examples

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

  Get the support:

      >>> a = 0.659
      >>> lb, ub = powerlaw.support(a)

  Calculate the first four moments:

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

  Display the probability density function (`pdf`):

      >>> x = np.linspace(powerlaw.ppf(0.01, a),
      ...                 powerlaw.ppf(0.99, a), 100)
      >>> ax.plot(x, powerlaw.pdf(x, a),
      ...        'r-', lw=5, alpha=0.6, label='powerlaw pdf')

  Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

  Freeze the distribution and display the frozen `pdf`:

      >>> rv = powerlaw(a)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = powerlaw.ppf([0.001, 0.5, 0.999], a)
      >>> np.allclose([0.001, 0.5, 0.999], powerlaw.cdf(vals, a))
      True

  Generate random numbers:

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

  And compare the histogram:

      >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
      >>> ax.set_xlim([x[0], x[-1]])
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()