Draw samples from a Weibull distribution.

This docstring was copied from numpy.random.mtrand.RandomState.weibull.

Some inconsistencies with the Dask version may exist.

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 $$X = \lambda(-ln(U))^{1/a}$$.

Note

New code should use the weibull method of a default_rng() instance instead; please see the Quick Start.

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.

scipy.stats.weibull_max
scipy.stats.weibull_min
scipy.stats.genextreme
gumbel
random.Generator.weibull

which should be used for new code.

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:

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


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

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)

>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()