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 : 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
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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
[R179] | 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. |
[R180] | 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)