The alias and alias-rejection-mixture methods for generating random variables from probability distributions

The alias-rejection-mixture method is a general and exact method for the computer generation of random variables from an arbitrary discrete, continuous, or mixed probability distribution. The method is based on two ingredients: (1) Walker's alias method, an ingenious and efficient method for generating discrete random variables, and (2) the rejection-mixture method, which is a modification of the standard rejection method that eliminates the need to repeat steps of the algorithm. The generation of random variables by the alias-rejection-mixture method requires operations that are simple and few and, remarkably, approximately the same for all distributions.