PlanetMath: Weibull random variable

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (17)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Weibull random variable

(Definition)

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

$\displaystyle f_X(x)=\frac{\gamma}{\alpha}(\frac{x-\mu}{\alpha})^{\gamma-1} e^{-(\frac{x-\mu}{\alpha})^\gamma}$

where $\alpha,\gamma,\mu\in\mathbb{R}$ , $\alpha,\gamma>0$ and $x\ge\mu$ . $\alpha$ is the scale parameter, $\gamma$ is the shape parameter, and $\mu$ is the location parameter.

Notation for having a Weibull distribution is $X\sim$ Wei $(\alpha,\gamma,\mu)$ . Usually, the location and scale parameters are dropped by the transformation

$\displaystyle Y=\frac{X-\mu}{\alpha}$

so that $Y\sim$ Wei $(\gamma):=$ Wei $(1,\gamma,0)$ . The resulting distribution is called the standard Weibull, or Rayleigh distribution:

$\displaystyle f_X(x)=\gamma x^{\gamma-1}\operatorname{exp}(-x^\gamma)$

Properties: Given a standard Weibull distribution $X\sim$ Wei $(\gamma)$ :

$\operatorname{E}[X]=\Gamma(\frac{\gamma+1}{\gamma})$ , where $\Gamma$ is the gamma function
Median = $(\operatorname{ln}2)^{\frac{1}{\gamma}}$
Mode $= \begin{cases} (1-\frac{1}{\gamma})^{1/\gamma} & \mbox{if $\gamma>1$}\ 0 & \mbox{otherwise} \end{cases}$
$\operatorname{Var}[X]=\Gamma(\frac{\gamma+2}{\gamma})-\Gamma(\frac{\gamma+1}{\gamma})^2$
$X\sim$ Wei $(\alpha,\gamma,0)$ iff $X^{\gamma}\sim$ Exp $(\alpha^\gamma)$ , the exponential distribution with parameter $\alpha^\gamma$

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

"Weibull random variable" is owned by CWoo.

(view preamble)

Other names:

Weibull distribution, Rayleigh distribution

Log in to rate this entry.
(view current ratings)

Cross-references: exponential distribution, iff, mode, median, gamma function, distribution, transformation, parameter, scale parameter, probability density function
There are 4 references to this entry.

This is version 5 of Weibull random variable, born on 2004-06-24, modified 2007-06-06.
Object id is 5960, canonical name is WeibullRandomVariable.
Accessed 11569 times total.

Classification:

AMS MSC:	60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
	62E15 (Statistics :: Distribution theory :: Exact distribution theory)
	62N99 (Statistics :: Survival analysis and censored data :: Miscellaneous)
	62P05 (Statistics :: Applications :: Applications to actuarial sciences and financial mathematics)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact