ABSTRACT
Given a random variable X defined on a probability space, Chapter 4 of Book II derived the theoretical basis for, and several constructions of a probability space on which could be defined a countable collection of independent random variables, identically distributed as X. Such spaces provide a rigorous framework for the laws of large numbers of that book, and the limit theorems of this book’s Chapter 6. This framework is in the background here, but the focus is on the actual generation of random sample collections using the previous theory and the various distribution functions introduced in previous chapters. The various sections then exemplify simulation approaches for discrete distributions, and then continuous distributions, using the left-continuous inverse function F*(y) and independent, continuous uniform variates commonly provided by various mathematical software. For generating normal, lognormal and Student-T variates, these constructions are at best approximate, and the chapter derives the exact constructions underlying the Box-Muller transform and the Bailey transform, respectively. The final section turns to the simulation of order statistics, both directly and with the aid of the Rényi representation theorem.
