Past exam of the simulation and statistical programming course of the University of Oxford / 2021 / 2 / a / i

For the exponential distribution with rate $\lambda$, the cumulative distribution function is $F(x)=1-e^{-\lambda x}$ for $x\ge 0$, and zero for $x<0$. Its quantile function isFor inverse transform sampling, draw $U$ from the uniform distribution on $(0,1)$ and return $-\log (1-U)/\lambda$. Equivalently return $-\log U/\lambda$, since $1-U$ has the same uniform distribution. Indeed, for $x\ge 0$, $\mathbb{P}(-\log U/\lambda >x)=\mathbb{P}(U<e^{-\lambda x})=e^{-\lambda x}$.