Weibull distribution

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Story

The quantity $y^{\alpha }$ is Weibull distributed if $y$ is Exponentially distributed. Like the Exponential distribution, it describes waiting times for arrival of a process. For $\alpha >1$, the longer we have waited, the more likely the event is to arrive, and vice versa for $\alpha <1$.

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Example

This is a model for aging. The longer an organism lives, the more likely it is to die.

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Parameters

There are two parameters, both strictly positive: the shape parameter $\alpha$, which dictates the shape of the curve, and the scale parameter $\sigma$, which dictates the rate of arrivals of the modeled process.

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Support

The Weibull distribution has support on the nonnegative real numbers.

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Probability density function

$$\begin{array}{r}f(y;\alpha ,\sigma )=\frac{\alpha }{\sigma }{\left(\frac{y}{\sigma }\right)}^{\alpha -1}{\mathrm{e}}^{-(y/\sigma {)}^{\alpha }}.\end{array}$$

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Cumulative distribution function

$$\begin{array}{r}F(y;\alpha ,\sigma )=1-{\mathrm{e}}^{-(x/\sigma {)}^{\alpha }}.\end{array}$$

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Moments

Mean: ${\displaystyle \sigma \mathrm{\Gamma }(1+1/\alpha )}$

Variance: ${\displaystyle {\sigma }^2[\mathrm{\Gamma }(1+2/\alpha )-{(\mathrm{\Gamma }(1+1/\alpha ))}^2]}$

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Usage

Package	Syntax
NumPy	sigma * rng.weibull(alpha)
SciPy	scipy.stats.weibull_min(alpha, loc=0, scale=sigma)
Distributions.jl	Weibull(alpha, sigma)
Stan	weibull(alpha, sigma)

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Related distributions

• The special case where $\alpha =1$ is the Exponential distribution with rate parameter $\beta =1/\sigma$.

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Notes

• SciPy has a location parameter, which should be set to zero, with $\sigma$ being the scale parameter.

• NumPy only provides a version of the Weibull distribution with $\sigma =1$. Sampling out of the Weibull distribution may be accomplished by multiplying the resulting samples by $\sigma$.

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PDF and CDF plots

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Links

• Wikipedia

• Numpy

• Scipy

• Distributions.jl

• Stan