A random variable \(T\) has the \(\text{Weibull}(\lambda,\alpha)\) if it has probability density function
\[f(t)=\lambda \alpha t^{\alpha-1} e^{-\lambda t^\alpha} \qquad (t>0)\]
where \(\lambda >0\) and \(\alpha>0\).
- Show that \(T^\alpha\) has an \(\text{exponential}(\lambda)\) distribution.
- Show that if \(U\) is a \(\text{uniform}(0,1)\) random variable, then
\[ T=\Big( – \frac{\log(U)}{\lambda}\Big)^{\frac1\alpha}\]
has a \(\text{Weibull}(\lambda,\alpha)\) distribution.