Weibull distribution
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Template:Probability distribution
In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function
- <math> f(x|k,\lambda) = (k/\lambda) (x/\lambda)^{(k-1)} e^{-(x/\lambda)^k}\,<math>
where <math>x >0<math> and <math>k >0<math> is the shape parameter and <math>\lambda >0<math> is the scale parameter of the distribution.
The cumulative density function is defined as
- <math>F(x|k,\lambda) = 1- e^{-(x/\lambda)^k}\,<math>
where again, <math>x >0<math>. 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 <math>k<1<math> (resulting in a decreasing density <math>f<math>). If the failure rate of the device is constant over time, one chooses <math>k=1<math>, again resulting in a decreasing function <math>f<math>. If the failure rate of the device increases over time, one chooses <math>k>1<math> and obtains a density <math>f<math> 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.
The expected value and standard deviation of a Weibull random variable can be expressed in terms of the Gamma function:
- <math>\textrm{E}(X) = \lambda \Gamma(1+1/k)\,<math>
and
- <math>\textrm{var}(X) = \lambda^2[\Gamma(1+2/k) - \Gamma^2(1+1/k)]\,<math>
Related distributions
- <math>X \sim \mathrm{Exponential}(\lambda)<math> is an exponential distribution if <math>X \sim \mathrm{Weibull}(\gamma = 1, \lambda)<math>.
- <math>X \sim \mathrm{Rayleigh}(\beta)<math> is a Rayleigh distribution if <math>X \sim \mathrm{Weibull}(\gamma = 2, \beta)<math>.
External links
- The Weibull distribution (with examples, properties, and calculators). (http://www.xycoon.com/Weibull.htm)
- The Weibull plot. (http://www.itl.nist.gov/div898/handbook/eda/section3/weibplot.htm)sv:Weibullfördelning