Simulation Generation Of Non - Uniform Random Numbers

Problem: Given a uniform random variable U, generate a random variable X having a prescribed distribution function F X) (⋅. We previously discussed the inversion method. While inversion is a very general method, it may be computationally expensive. In particular, computing 1 X F () − ⋅ may have to be implemented via a numerical root-finding method in many cases. Therefore, we will now describe other methods for non-uniform random number generation. 1. Acceptance-Rejection Method We now discuss a method that is well-suited to generating random variates with an easily-calculated density. This method is called acceptance-rejection. Goal: Generate a random variate X having given probability density f X (x), where f X (x) is positive only on the interval [a, b] (where < a < b < ∞-∞). Enclose the density in a rectangle having base (b−a) and height m, where m = sup f (x) a x b X ≤ ≤. f X (x) a b x m Suppose we throw down points uniformly in the rectangle R (denote these points by the symbol x). Throw away (or reject) the points above the density f X (x). Claim: The x-coordinate of each accepted point (denoted by the symbol ⊗) has density f X (x). Proof: Let be the (x, y) coordinates of a random point distributed uniformly in R. Then, for a x ≤ b,