Basic Principles: Rejection, Weighting, and Others

Abstract

To generate random variables that follow a general probability distribution function 1r, we need first to generate random variables uniformly distributed in [0,1]. These random variables are often called random numbers for simplicity. However, this "simple-sounding" task is not easily achievable on a computer. But even if it were possible, it might not be desirable to use authentic random numbers because of the need to debug computer programs. In debugging a program, we often have to repeat the same computation many times; this require us to reproduce the same sequence of random numbers repeatedly. What becomes an accepted alternative in the community of scientific computing is to generate pseudo-random numbers. More formally, we can define a uniform pseudo-random number generator as an algorithm which, starting from an initial value u0 (i.e., the seed), produces a sequence (u;) = (Di(u0)) of values in [0,1]. For all n, the values (u1 , ... , un) should reproduce the behavior of an i.i.d. sample (V1 , ... , Vn) of uniform random variables. A few very good pseudo-random number generators are available; we refer the reader to Marsaglia and Zaman (1993) and Knuth (1997) for further reference. Consequently, we assume from now on that uniform random variables can be satisfactorily produced on the computer. The following simple lemma enables us to produce nonuniform random variables. Its proof is left as an exercise for the reader.