4.5.1 The Weibull Distribution

The family of Weibull distributions was introduced by the Swedish physicist Waloddi Weibull in 1939; his 1951 article “A Statistical Distribution Function of Wide Applicability” (J. of Applied Mechanics, vol. 18: 293-297) discusses a number of applications.

Weibull distribution

A random variable $X$ is said to have a Weibull distribution with shape parameter $\alpha$ and scale parameter $\beta \left(\alpha >0,\beta >0\right)$ if the pdf of $X$ is

$$f\left(x;\alpha ,\beta \right)=\{\begin{array}{cc}\frac{\alpha }{{\beta }^{\alpha }}{x}^{\alpha -1}{e}^{-{\left(x/\beta \right)}^{\alpha }}& x\ge 0\\ 0& x<0\end{array}\text{\hspace{1em}(4.11)}$$

In some situations, there are theoretical justifications for the appropriateness of the Weibull distribution, but in many applications $f\left(x;\alpha ,\beta \right)$ simply provides a good fit to observed data for particular values of $\alpha$ and $\beta$. When $\alpha =1$, the pdf reduces to the exponential distribution (with $\lambda =1/\beta$ ), so the exponential distribution is a special case of both the gamma and Weibull distributions. However, there are gamma distributions that are not Weibull distributions and vice versa, so one family is not a subset of the other. Both $\alpha$ and $\beta$ can be varied to obtain a number of different-looking density curves, as illustrated in Figure 4.28.

Figure 4.28 Weibull density curves

Integrating to obtain $E\left(X\right)$ and $E\left(X^2\right)$ yields

\mu = {\beta \Gamma }\left( {1 + \frac{1}{\alpha }}\right)$$ $${\sigma }^{2} = {\beta }^{2}\left\{ {\Gamma \left( {1 + \frac{2}{\alpha }}\right) - {\left\lbrack \Gamma \left( 1 + \frac{1}{\alpha }\right) \right\rbrack }^{2}}\right\}

The computation of $\mu$ and ${\sigma }^2$ thus necessitates using the gamma function.

The integration ${\int }_0^{x}f\left(y;\alpha ,\beta \right)dy$ is easily carried out to obtain the cdf of $X$ .

cdf of Weibull rv

The cdf of a Weibull rv having parameters $\alpha$ and $\beta$ is

$$F\left(x;\alpha ,\beta \right)=\{\begin{array}{cc}0& x<0\\ 1-e^{-{\left(x/\beta \right)}^{\alpha }}& x\ge 0\end{array}\text{\hspace{1em}(4.12)}$$

EX 4.25

In practical situations, a Weibull model may be reasonable except that the smallest possible $X$ value may be some value $\gamma$ not assumed to be zero (this would also apply to a gamma model; see Exercise 66). The quantity $\gamma$ can then be regarded as a third (threshold or location) parameter of the distribution, which is what Weibull did in his original work. For, say, $\gamma =3$ , all curves in Figure 4.28 would be shifted 3 units to the right. This is equivalent to saying that $X-\gamma$ has the pdf (4.11), so that the cdf of $X$ is obtained by replacing $x$ in (4.12) by $x-\gamma$ .

EX 4.26