Skip to main content
`Ctrl`+`K`

# scipy.stats.exponnorm

scipy.stats.exponnorm = <scipy.stats._continuous_distns.exponnorm_gen object>

  An exponentially modified Normal continuous random variable.

  Also known as the exponentially modified Gaussian distribution [1].

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

  Notes

  The probability density function for `exponnorm` is:

  \[f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right) \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)\]

  where \(x\) is a real number and \(K > 0\).

  It can be thought of as the sum of a standard normal random variable and an independent exponentially distributed random variable with rate `1/K`.

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

  An alternative parameterization of this distribution (for example, in the Wikipedia article [1]) involves three parameters, \(\mu\), \(\lambda\) and \(\sigma\).

  In the present parameterization this corresponds to having `loc` and `scale` equal to \(\mu\) and \(\sigma\), respectively, and shape parameter \(K = 1/(\sigma\lambda)\).

  Added in version 0.16.0.

  References

  [1] (1,2)

  Exponentially modified Gaussian distribution, Wikipedia, https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution

  Examples

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

  Get the support:

      >>> K = 1.5
      >>> lb, ub = exponnorm.support(K)

  Calculate the first four moments:

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

  Display the probability density function (`pdf`):

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

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

      >>> r = exponnorm.rvs(K, 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()