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# scipy.stats.gennorm

scipy.stats.gennorm = <scipy.stats._continuous_distns.gennorm_gen object>

  A (symmetric) generalized normal continuous random variable.

  As an instance of the `rv_continuous` class, `gennorm` 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(beta, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, beta, loc=0, scale=1) | Probability density function.
  logpdf(x, beta, loc=0, scale=1) | Log of the probability density function.
  cdf(x, beta, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, beta, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, beta, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, beta, loc=0, scale=1) | Log of the survival function.
  ppf(q, beta, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, beta, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, beta, loc=0, scale=1) | Non-central moment of the specified order.
  stats(beta, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(beta, 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=(beta,), 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(beta, loc=0, scale=1) | Median of the distribution.
  mean(beta, loc=0, scale=1) | Mean of the distribution.
  var(beta, loc=0, scale=1) | Variance of the distribution.
  std(beta, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, beta, loc=0, scale=1) | Confidence interval with equal areas around the median.

  See also

  `laplace`

    Laplace distribution

  `norm`

    normal distribution

  Notes

  The (symmetric) generalized normal distribution is also known as the Subbotin distribution, exponential power distribution, and generalized error distribution [1].

  The probability density function for `gennorm` is [1]:

  \[f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),\]

  where \(x\) is a real number, \(\beta > 0\) and \(\Gamma\) is the gamma function (`scipy.special.gamma`).

  `gennorm` takes `beta` as a shape parameter for \(\beta\). For \(\beta = 1\), it is identical to a Laplace distribution. For \(\beta = 2\), it is identical to a normal distribution (with `scale=1/sqrt(2)`).

  References

  [1] (1,2)

  “Generalized normal distribution, Version 1”, https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

  [2]

  Nardon, Martina, and Paolo Pianca. “Simulation techniques for generalized Gaussian densities.” Journal of Statistical Computation and Simulation 79.11 (2009): 1317-1329

  [3]

  Wicklin, Rick. “Simulate data from a generalized Gaussian distribution” in The DO Loop blog, September 21, 2016, https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html

  Examples

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

  Get the support:

      >>> beta = 1.3
      >>> lb, ub = gennorm.support(beta)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = gennorm.stats(beta, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(gennorm.ppf(0.01, beta),
      ...                 gennorm.ppf(0.99, beta), 100)
      >>> ax.plot(x, gennorm.pdf(x, beta),
      ...        'r-', lw=5, alpha=0.6, label='gennorm 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 = gennorm(beta)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

      >>> r = gennorm.rvs(beta, 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()