# Weibull distribution

The Weibull distribution is a continuous probability distribution which is bounded below and unbounded above. It has three parameters$a$, location; $$b>0$$, scale; $$c>0$$, shape. Its probability density function is given by

$$\frac{c}{b}\left(\frac{x-a}{b}\right)^{c-1}\exp\left(-\left(\frac{x-a}{b}\right)^{c}\right),\;\;\;x>a$$.

The Weibull distribution is useful for modelling extreme values, representing the distribution of the maximum value out of a large number of random variables drawn from a bounded distribution.

Mean = $$a+b\Gamma\left(\frac{c+1}{c}\right)$$ and Variance = $$b^2\left(\Gamma\left(\frac{c+2}{c}\right)-\Gamma\left(\frac{c+1}{c}\right)^2\right)$$

where $$\Gamma(\cdot)$$ is the Gamma function.

## References

• Spiegel, MR, Schiller, JJ, Srinivasan, RA 2000 Schaum's Outline of Probability and Statistics. McGraw-Hill