}{k^{\frac{1}{k}}}, if $k>1$| variance =$\backslash lambda^2\backslash Gamma\backslash left(1+\backslash frac\{2\}\{k\}\backslash right)\; -\; \backslash mu^2\backslash ,$| skewness =$\backslash frac\{\backslash Gamma(1+\backslash frac\{3\}\{k\})\backslash lambda^3-3\backslash mu\backslash sigma^2-\backslash mu^3\}\{\backslash sigma^3\}$| kurtosis =(see text)| entropy =$\backslash gamma\backslash left(1\backslash !-\backslash !\backslash frac\{1\}\{k\}\backslash right)+\backslash ln\backslash left(\backslash frac\{\backslash lambda\}\{k\}\backslash right)+1$| mgf = see Weibull fading| char =| }}

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

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

for $x\; \backslash geq\; 0$ and f(x; k, λ) = 0 for x < 0, where $k\; >0$ is the shape parameter and $\backslash lambda\; >0$ is the scale parameter of the distribution.

The cumulative distribution function is

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

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

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

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

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

Properties

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The nth raw moment is given by:

$m\_n\; =\; \backslash lambda^n\; \backslash Gamma(1+n/k)\backslash ,$

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

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

and

$\backslash textrm\{var\}(X)\; =\; \backslash lambda^2-\; \backslash Gamma^2(1+1/k)\backslash ,$

The skewness is given by:

$\backslash gamma\_1=\backslash frac\{\backslash Gamma\backslash left(1+\backslash frac\{3\}\{k\}\backslash right)\backslash lambda^3-3\backslash mu\backslash sigma^2-\backslash mu^3\}\{\backslash sigma^3\}$

The kurtosis excess is given by:

$\backslash gamma\_2=\backslash frac\{-6\backslash Gamma\_1^4+12\backslash Gamma\_1^2\backslash Gamma\_2-3\backslash Gamma\_2^2$

-4\Gamma_1\Gamma_3+\Gamma_4}{^*^2}

where $\backslash Gamma\_i=\backslash Gamma(1+i/k)$. The kurtosis excess may also be written as

$\backslash gamma\_2=\backslash frac\{\backslash lambda^4\backslash Gamma\backslash left(1+\backslash frac\{4\}\{k\}\backslash right)$

-3\sigma^4-4\gamma_1\sigma^3\mu-6\sigma^2\mu^2-\mu^4}{\sigma^4}.

Generating Weibull-distributed random variates

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

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

has a Weibull distribution with parameters k and λ. This follows from the form of the cumulative distribution function.

Related distributions

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

Uses

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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.

External links

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• The Weibull distribution (with examples, properties, and calculators).
• The Weibull plot.
• Weibull plotting paper.
• Using Excel for Weibull Analysis
  This article from the Quality Digest describes how to use MS Excel to analyse lifetest data with the Weibull statistical distribution. Although Excel doesn't have an inverse Weibull function, this article shows how to use Excel to solve for critical values.

Continuous distributions

Weibull-Verteilung | Distribution de Weibull | Distribución Weibull | Variabile casuale di Weibull | Weibull-verdeling | Распределение Вейбулла | Sebaran Weibull | Weibullfördelning