numpy.random.Generator.weibull

method

random.Generator.weibull(a, size=None)

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

Parameters:

afloat or array_like of floats

Shape parameter of the distribution. Must be nonnegative.

sizeint or tuple of ints, optional

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

Returns:

outndarray or scalar

Drawn samples from the parameterized Weibull distribution.

See also

scipy.stats.weibull_max

scipy.stats.weibull_min

scipy.stats.genextreme

gumbel

Notes

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

When a = 1, the Weibull distribution reduces to the exponential distribution.

References

[1]

Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.

[2]

Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.

[3]

Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution

Examples

Draw samples from the distribution:

>>> rng = np.random.default_rng()
>>> a = 5. # shape
>>> s = rng.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> def weibull(x, n, a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
>>> count, bins, _ = plt.hist(rng.weibull(5., 1000))
>>> x = np.linspace(0, 2, 1000)
>>> bin_spacing = np.mean(np.diff(bins))
>>> plt.plot(x, weibull(x, 1., 5.) * bin_spacing * s.size, label='Weibull PDF')
>>> plt.legend()
>>> plt.show()