 Solution

For $\alpha ,\beta >0$, integrating the Weibull distribution probability density function givesThe inverse transform sampling method takes $U$ uniform on $(0,1)$ and solves $G(X)=U$. HenceSince $1-U$ is also uniform, $\{-\beta \log U{\}}^{1/\alpha }$ is an equivalent implementation. For any $x\ge 0$, the resulting event $X\le x$ has probability $G(x)$, verifying its distribution.