Random Number Generation 1

Universit e de Montr eal 2 4.1 INTRODUCTION Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0; 1) random variables (i.e., continuous random variables distributed uniformly over the interval (0; 1)). These random numbers are then transformed as needed to simulate random variables from diierent probability distributions, such as the normal, exponential, Pois-son, binomial, geometric, discrete uniform, etc., as well as multivariate distributions and more complicated random objects. In general, the validity of the transformation methods depends strongly on the IID U(0; 1) assumption. But this assumption is false, since the random number generators are actually simple deterministic programs trying to fool the user by producing a deterministic sequence that looks random. What could be the impact of this on the simulation results? Despite this problem, are there \safe" generators? What about the generators commonly available in system libraries and simulation packages? If they are not satisfactory, how can we build better ones? Which ones should be used, and where is the code? These are some of the topics addressed in this chapter.