Weibull random variable

X is a Weibull random variable if it has a probability density functionMathworldPlanetmath, given by


where α,γ,μ, α,γ>0 and xμ. α is the scale parameter, γ is the shape parameter, and μ is the location parameter.

Notation for X having a Weibull distribution is XWei(α,γ,μ). Usually, the location and scale parameters are dropped by the transformation


so that YWei(γ):=Wei(1,γ,0). The resulting distributionDlmfPlanetmath is called the standard Weibull, or Rayleigh distribution:


: Given a standard Weibull distribution XWei(γ):

  1. 1.

    E[X]=Γ(γ+1γ), where Γ is the gamma functionDlmfDlmfMathworldPlanetmath

  2. 2.

    Median = (ln2)1γ

  3. 3.

    Mode ={(1-1γ)1/γif γ>10otherwise

  4. 4.


  5. 5.

    XWei(α,γ,0) iff XγExp(αγ), the exponential distributionMathworldPlanetmath with parameterMathworldPlanetmath αγ

Remark. The Weibull distribution is often used to model reliability or lifetime of such as light bulbs.

Title Weibull random variable
Canonical name WeibullRandomVariable
Date of creation 2013-03-22 14:26:44
Last modified on 2013-03-22 14:26:44
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 62N99
Classification msc 62E15
Classification msc 60E05
Classification msc 62P05
Synonym Weibull distribution
Synonym Rayleigh distribution