numpy.random.weibull

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

Weibull distribution.

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

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}.

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.

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
probability density function, distribution or cumulative density function, etc.

gumbel, scipy.stats.distributions.genextreme

Notes

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

[95]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.
[96]Waloddi Weibull, 1951 “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper.
[97]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
>>> def weib(x,n,a):
...     return (a/n)*(x/n)**(a-1)*exp(-(x/n)**a)
>>> count, bins, ignored = plt.hist(numpy.random.weibull(5.,1000))
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> x = arange(1,100.)/50.
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()

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