3.6 The Weibull Distribution: $\mathrm{𝖶𝖾𝗂𝖻}(\alpha ,\beta )$

Parameters: $𝜽=(\alpha ,\beta )$ with a shape parameter $\alpha >0$ and a rate parameter $\beta >0$.

1. $f_X(x;𝜽)=\alpha {\beta }^{\alpha }{x}^{\alpha -1}\exp \{-{(\beta x)}^{\alpha }\}$ for ,

2. $F_X(x)=1-\exp (-{(\beta x)}^{\alpha })$,

3. $𝖤\left[X\right]=\mathrm{\Gamma }(1+{\alpha }^{-1})/\beta$,

4. $\mathrm{𝖵𝖺𝗋}\left[X\right]=\{\mathrm{\Gamma }(1+2{\alpha }^{-1})-\mathrm{\Gamma }{(1+{\alpha }^{-1})}^2\}/{\beta }^2$.

We write $X\sim \mathrm{Weibull}(\alpha ,\beta )$.

Other: The $\mathrm{Weibull}(1,\beta )$ distribution is the $\mathrm{Exponential}(\beta )$ distribution. If $Y\sim \mathrm{𝖤𝗑𝗉}(1)$ then $X=Y^{(1/\alpha )}\sim \mathrm{𝖶𝖾𝗂𝖻}(\alpha ,1)$.

Usage: The Weibull distribution is a generalisation of the exponential distribution and, like the exponential distribution, is often used to model failure times such as the lifetime of a part, a battery, or a person.

Unnumbered Figure: First link, Second Link

If $T\sim \mathrm{𝖤𝗑𝗉}(\beta )$ then by the memoryless property, for small $\mathrm{\Delta }t$,

In any fixed small time interval, the probability of failure in that interval given survival up to just before the interval is proportional to the width of that interval and independent of when that interval occurs.

But a part might get worn out and as it gets older and might be more likely to fail tomorrow given that it has survived up until today. Or a person alone in the jungle who has survived for 2 years might be more likely to survive tomorrow than someone who has just been dropped into the jungle, because the first person has clearly learned to adapt. The Weibull allows for these two possibilities.