Weibull distribution

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**Weibull**
Probability density function
Cumulative distribution function
Parameters	$\lambda>0\,$ scale (real) $k>0\,$ shape (real)
Support	$x \in [0; +\infty)\,$
Template:Probability distribution/link density	$(k/\lambda) (x/\lambda)^{(k-1)} e^{-(x/\lambda)^k}$
cdf	$1- e^{-(x/\lambda)^k}$
Mean	$\mu=\lambda \Gamma(1+1/k)\,$
Median	$\lambda\ln(2)^{1/k}\,$
Mode
Variance	$\sigma^2=\lambda^2\Gamma(1+2/k) - \mu^2\,$
Skewness	$\frac{\Gamma(1+3/k)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}$
Kurtosis
Entropy	$\gamma\left(1\!-\!\frac{1}{k}\right)+\left(\frac{\lambda}{k}\right)^k +\ln\left(\frac{\lambda}{k}\right)$
mgf
Char. func.

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

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

where $x \geq0$ and $k > 0$ is the shape parameter and $λ > 0$ is the scale parameter of the distribution.

The cumulative density function is defined as

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

where again, $x > 0$ . Weibull distributions are often used to model the time until a given technical device fails. If the failure rate of the device decreases over time, one chooses $k < 1$ (resulting in a decreasing density $f$ ). If the failure rate of the device is constant over time, one chooses $k = 1$ , again resulting in a decreasing function $f$ . If the failure rate of the device increases over time, one chooses $k > 1$ and obtains a density $f$ which increases towards a maximum and then decreases forever. Manufacturers will often supply the shape and scale parameters for the lifetime distribution of a particular device. The Weibull distribution can also be used to model the distribution of wind speeds at a given location on Earth. Again, every location is characterized by a particular shape and scale parameter.

1 Properties
2 Generating Weibull-distributed random variates
3 Related distributions
4 Uses
5 External links

Properties

The n-th raw moment is given by:

$m_n = \lambda^n \Gamma(1+n/k)\,$

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

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

and

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

Generating Weibull-distributed random variates

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

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

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

Related distributions

$X ˜Exponential(λ)$ is an exponential distribution if $X ˜Weibull(γ = 1,λ)$ .
$X ˜Rayleigh(β)$ is a Rayleigh distribution if $X ˜Weibull(γ = 2,β)$ .
$\lambda(-\ln(X))^{1/k}\,$ is a Weibull distribution if $X ˜Uniform(0,1)$ .
See also the generalized extreme value distribution.

Uses

The Weibull distribution gives the distribution of lifetimes of objects.It is also used in analysis of systems involving a weakest link.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. And it is very important in Extreme value theory.

External links

The Weibull distribution (with examples, properties, and calculators).
The Weibull plot.
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.
WeibPar.com This site offers standalone software to estimate the Weibull parameters from a given set of data. Although it was developed to characterize Weibull parameters from failure data for a material, it can be used to calculate Weibull parameters for any set of data.
Weibull.com Another site offering Weibull information.de:Weibull-Verteilung

nl:Weibull-verdeling su:Sebaran Weibull sv:Weibullfördelning

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Category: Continuous distributions