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# scipy.stats.moyal

scipy.stats.moyal = <scipy.stats._continuous_distns.moyal_gen object>

  A Moyal continuous random variable.

  As an instance of the `rv_continuous` class, `moyal` 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.

  Notes

  The probability density function for `moyal` is:

  \[f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}\]

  for a real number \(x\).

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `moyal.pdf(x, loc, scale)` is identically equivalent to `moyal.pdf(y) / 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.

  This distribution has utility in high-energy physics and radiation detection. It describes the energy loss of a charged relativistic particle due to ionization of the medium [1]. It also provides an approximation for the Landau distribution. For an in depth description see [2]. For additional description, see [3].

  References

  [1]

  J.E. Moyal, “XXX. Theory of ionization fluctuations”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol 46, 263-280, (1955). DOI:10.1080/14786440308521076 (gated)

  [2]

  G. Cordeiro et al., “The beta Moyal: a useful skew distribution”, International Journal of Research and Reviews in Applied Sciences, vol 10, 171-192, (2012). http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf

  [3]

  C. Walck, “Handbook on Statistical Distributions for Experimentalists; International Report SUF-PFY/96-01”, Chapter 26, University of Stockholm: Stockholm, Sweden, (2007). http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

  New in version 1.1.0.

  Examples

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

  Calculate the first four moments:

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

  Display the probability density function (`pdf`):

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

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

      >>> r = moyal.rvs(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()

  Methods

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