Weibull distribution.
Draw samples from a 1-parameter Weibull distribution with the given
shape parameter 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
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is just
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Parameters: | a : float
Shape of the distribution.
size : int or tuple of ints, optional
Output shape. If the given shape is, e.g., (m, n, k), then
m * n * k samples are drawn. Default is None, in which case a
single value is returned.
|
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
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{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
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where
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is the shape and
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the scale.
The function has its peak (the mode) at
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.
When a = 1, the Weibull distribution reduces to the exponential
distribution.
References
[R240] | 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. |
[R241] | Waloddi Weibull, 1951 “A Statistical Distribution Function of Wide
Applicability”, Journal Of Applied Mechanics ASME Paper. |
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)