Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 4 / a / Solution

For an invertible continuous cumulative distribution function, monotonicity givesThus each transformed observation has the desired distribution. More generally use the quantile function $q(u)=\inf \{x:F(x)\ge u\}$; the same event identity proves inverse transform sampling even when there are atoms. Independent uniform inputs give independent transformed samples; mere uniform marginal distributions do not imply independence between different samples.