SciPy

numpy.random.weibull

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

Weibull distribution.

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

System Message: WARNING/2 (X = (-ln(U))^{1/a} )

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

The more common 2-parameter Weibull, including a scale parameter

System Message: WARNING/2 (\lambda)

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is just

System Message: WARNING/2 (X = \lambda(-ln(U))^{1/a})

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2015/01/01> patch level 2 Babel <3.9m> and hyphenation patterns for 79 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.173 \endinput ^^M No pages of output. Transcript written on math.log.
.

Parameters:

a : float

Shape of the distribution.

size : tuple of ints

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn.

See also

scipy.stats.distributions.weibull_max, scipy.stats.distributions.weibull_min, scipy.stats.distributions.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

System Message: WARNING/2 (p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, )

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where

System Message: WARNING/2 (a)

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is the shape and

System Message: WARNING/2 (\lambda)

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the scale.

The function has its peak (the mode) at

System Message: WARNING/2 (\lambda(\frac{a-1}{a})^{1/a})

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/TeX Live for SUSE Linux) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2015/01/01> patch level 2 Babel <3.9m> and hyphenation patterns for 79 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.173 \endinput ^^M No pages of output. Transcript written on math.log.
.

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

References

[R238]Waloddi Weibull, Professor, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.
[R239]Waloddi Weibull, 1951 “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper.
[R240]Wikipedia, “Weibull distribution”, http://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()

(Source code, png, pdf)

../../_images/numpy-random-weibull-1.png

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