## CHAPTER 2

## Pseudo-Random Numbers

Almost all the simulation methods and algorithms to be discussed in later chapters derive their randomness from an infinite supply *U*_{0}, *U*_{1}, *U*_{2}, . . . of *random numbers*; that is, an independent sequence (*U*_{i}) of random variables uniformly distributed on (0, 1). Many users of simulation are content to remain ignorant of how such numbers are produced, merely calling standard functions to produce them. Such attitudes are dangerous, for random numbers are the foundations of our simulation edifice, and problems at higher levels are frequently traced back to faulty foundations.

This chapter is of rather specialized appeal. Do not yield to the temptation to skip it without working exercise 2.17, for many of the random number generators in use (at the time of writing) have serious defects.

### 2.1. HISTORY AND PHILOSOPHY

There is no mathematical problem with random numbers: their existence is provable from Kolmogorov’s axioms for probability. [See, for example, Neveu (1965, Section 5.1).] However, this result does not produce a realization of a sequence of random numbers for us to use; we have to find some observable process of which the mathematics is a reasonable model. The philosophical problem hinges on that much-abused word “reasonable.” How can we decide from a finite sequence (*U*_{1}, . . . , *U*_{n}) whether *(U*_{i}) is an adequate model? We have immediately all the philosophical problems of statistical inference.

The earliest users of simulation used physical processes ...