# Weibull distribution

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## Overview

In probability theory and statistics, the Weibull distribution[1] (named after Waloddi Weibull) is a continuous probability distribution with the probability density function

${\displaystyle f(x;k,\lambda )={k \over \lambda }\left({x \over \lambda }\right)^{k-1}e^{-(x/\lambda )^{k}}\,}$

for ${\displaystyle x\geq 0}$ and f(x; k, λ) = 0 for x < 0, where ${\displaystyle k>0}$ is the shape parameter and ${\displaystyle \lambda >0}$ is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential.

The Weibull distribution is often used in the field of life data analysis due to its flexibility—it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then k < 1. If the failure rate is constant over time, then k = 1. If the failure rate increases over time, then k > 1.

An understanding of the failure rate may provide insight as to what is causing the failures:

• A decreasing failure rate would suggest "infant mortality". That is, defective items fail early and the failure rate decreases over time as they fall out of the population.
• A constant failure rate suggests that items are failing from random events.
• An increasing failure rate suggests "wear out" - parts are more likely to fail as time goes on.

When k = 3.4, then the Weibull distribution appears similar to the normal distribution. When k = 1, then the Weibull distribution reduces to the exponential distribution.

## Properties

The nth raw moment is given by:

${\displaystyle m_{n}=\lambda ^{n}\Gamma (1+n/k)\,}$

where ${\displaystyle \Gamma }$ is the Gamma function. The expected value and standard deviation of a Weibull random variable can be expressed as:

${\displaystyle {\textrm {E}}(X)=\lambda \Gamma (1+1/k)\,}$

and

${\displaystyle {\textrm {var}}(X)=\lambda ^{2}[\Gamma (1+2/k)-\Gamma ^{2}(1+1/k)]\,.}$

The skewness is given by:

${\displaystyle \gamma _{1}={\frac {\Gamma \left(1+{\frac {3}{k}}\right)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}.}$

The excess kurtosis is given by:

${\displaystyle \gamma _{2}={\frac {-6\Gamma _{1}^{4}+12\Gamma _{1}^{2}\Gamma _{2}-3\Gamma _{2}^{2}-4\Gamma _{1}\Gamma _{3}+\Gamma _{4}}{[\Gamma _{2}-\Gamma _{1}^{2}]^{2}}}}$

where ${\displaystyle \Gamma _{i}=\Gamma (1+i/k)}$. The kurtosis excess may also be written as

${\displaystyle \gamma _{2}={\frac {\lambda ^{4}\Gamma (1+{\frac {4}{k}})-4\gamma _{1}\sigma ^{3}\mu -6\mu ^{2}\sigma ^{2}-\mu ^{4}}{\sigma ^{4}}}}$

A generalized, 3-parameter Weibull distribution is also often found in the literature. It has the probability density function

${\displaystyle f(x;k,\lambda ,\theta )={k \over \lambda }\left({x-\theta \over \lambda }\right)^{k-1}e^{-({x-\theta \over \lambda })^{k}}\,}$

for ${\displaystyle x\geq \theta }$ and f(x; k, λ, θ) = 0 for x < θ, where ${\displaystyle k>0}$ is the shape parameter, ${\displaystyle \lambda >0}$ is the scale parameter and ${\displaystyle \theta }$ is the location parameter of the distribution. When θ=0, this reduces to the 2-parameter distribution.

The cumulative distribution function for the 2-parameter Weibull is

${\displaystyle F(x;k,\lambda )=1-e^{-(x/\lambda )^{k}}\,}$

for x ≥ 0, and F(x; k; λ) = 0 for x < 0.

The cumulative distribution function for the 3-parameter Weibull is

${\displaystyle F(x;k,\lambda ,\theta )=1-e^{-({x-\theta \over \lambda })^{k}}}$

for x ≥ θ, and F(x; k, λ, θ) = 0 for x < θ.

The failure rate h (or hazard rate) is given by

${\displaystyle h(x;k,\lambda )={k \over \lambda }\left({x \over \lambda }\right)^{k-1}.}$

## Generating Weibull-distributed random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

${\displaystyle X=\lambda (-\ln(U))^{1/k}\,}$

has a Weibull distribution with parameters k and λ. This follows from the form of the cumulative distribution function. Note that if you are generating random numbers belonging to (0,1), exclude zero values to avoid the natural log of zero.

## Related distributions

• ${\displaystyle X\sim \mathrm {Exponential} (\lambda )}$ is an exponential distribution if ${\displaystyle X\sim \mathrm {Weibull} (\gamma =1,\lambda ^{-1})}$.
• ${\displaystyle X\sim \mathrm {Rayleigh} (\beta )}$ is a Rayleigh distribution if ${\displaystyle X\sim \mathrm {Weibull} (\gamma =2,{\sqrt {2}}\beta )}$.
• ${\displaystyle \lambda (-\ln(X))^{1/k}\,}$ is a Weibull distribution if ${\displaystyle X\sim \mathrm {Uniform} (0,1)}$.
• Inverse Weibull distribution with p.d.f. ${\displaystyle f(x;k,\lambda )=(k/\lambda )(\lambda /x)^{(k+1)}e^{-(\lambda /x)^{k}}}$
• See also the generalized extreme value distribution.

## Uses

The Weibull distribution is most commonly used in life data analysis, though it has found other applications as well. The Weibull distribution is often used in place of the normal distribution due to the fact that a Weibull variate can be generated through inversion, while normal variates are typically generated using the more complicated Box-Muller method, which requires two uniform random variates. Weibull distributions may also be used to represent manufacturing and delivery times in industrial engineering problems, while it is very important in extreme value theory and weather forecasting. It is also a very popular statistical model in reliability engineering and failure analysis, while it is widely applied in radar systems to model the dispersion of the received signals level produced by some types of clutters. Furthermore, concerning wireless communications, the Weibull distribution may be used for fading channel modelling, since the Weibull fading model seems to exhibit good fit to experimental fading channel measurements. The Weibull distribution is also commonly used to describe wind speed distributions as the natural distribution often matches the Weibull shape.

## References

1. Weibull, W. (1951) "A statistical distribution function of wide applicability" J. Appl. Mech.-Trans. ASME 18(3), 293-297