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Change of variable: Weibull distribution

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 \(\alpha>0\).

  1. Show that \(T^\alpha\) has an \(\text{exponential}(\lambda)\) distribution.
  2. 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.

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