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# scipy.stats.weibull_min

scipy.stats.weibull_min = <scipy.stats._continuous_distns.weibull_min_gen object>

  Weibull minimum continuous random variable.

  The Weibull Minimum Extreme Value distribution, from extreme value theory (Fisher-Gnedenko theorem), is also often simply called the Weibull distribution. It arises as the limiting distribution of the rescaled minimum of iid random variables.

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

  See also

  `weibull_max`, `numpy.random.Generator.weibull`, `exponweib`

  Notes

  The probability density function for `weibull_min` is:

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

  for \(x > 0\), \(c > 0\).

  `weibull_min` takes `c` as a shape parameter for \(c\). (named \(k\) in Wikipedia article and \(a\) in `numpy.random.weibull`). Special shape values are \(c=1\) and \(c=2\) where Weibull distribution reduces to the `expon` and `rayleigh` distributions respectively.

  Suppose `X` is an exponentially distributed random variable with scale `s`. Then `Y = X**k` is `weibull_min` distributed with shape `c = 1/k` and scale `s**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, `weibull_min.pdf(x, c, loc, scale)` is identically equivalent to `weibull_min.pdf(y, c) / 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.

  References

  https://en.wikipedia.org/wiki/Weibull_distribution

  https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

  Examples

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

  Get the support:

      >>> c = 1.79
      >>> lb, ub = weibull_min.support(c)

  Calculate the first four moments:

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

  Display the probability density function (`pdf`):

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

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

      >>> r = weibull_min.rvs(c, 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()