Weibull distribution

**Weibull**
Probability density function Missing image Weibull1.png Missing image Weibull2.png
Cumulative distribution function
Parameters	$\lambda>0\,$ scale (real) $k>0\,$ shape (real)
Support	$x \in [0; +\infty)\,$
pdf	$(k/\lambda) (x/\lambda)^{(k-1)} e^{-(x/\lambda)^k}$
cdf	$1- e^{-(x/\lambda)^k}$
Mean	$\lambda \Gamma(1+1/k)\,$
Median
Mode
Variance	$\lambda^2[\Gamma(1+2/k) - \Gamma^2(1+1/k)]\,$
Skewness
Kurtosis
Entropy
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 $x > 0$ 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.

The expected value and standard deviation of a Weibull random variable can be expressed in terms of the Gamma function:

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

and

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

Related distributions

$X \sim \mathrm{Exponential}(\lambda)$ is an exponential distribution if $X \sim \mathrm{Weibull}(\gamma = 1, \lambda)$ .
$X \sim \mathrm{Rayleigh}(\beta)$ is a Rayleigh distribution if $X \sim \mathrm{Weibull}(\gamma = 2, \beta)$ .

Weibull distribution

Related distributions

External links

See also: Weibull distribution, Characteristic function, Cumulative distribution function, Expected value, Exponential distribution, Gamma function, Information entropy, Kurtosis