Page 92 - Computational Statistics Handbook with MATLAB

Chapter 4

Generating Random Variables

4.1 Introduction
Many of the methods in computational statistics require the ability to gener-
ate random variables from known probability distributions. This is at the
heart of Monte Carlo simulation for statistical inference (Chapter 6), boot-
strap and resampling methods (Chapters 6 and 7), Markov chain Monte
Carlo techniques (Chapter 11), and the analysis of spatial point processes
(Chapter 12). In addition, we use simulated random variables to explain
many other topics in this book, such as exploratory data analysis (Chapter 5),
density estimation (Chapter 8), and statistical pattern recognition
(Chapter 9).
There are many excellent books available that discuss techniques for gen-
erating random variables and the underlying theory; references will be pro-
vided in the last section. Our purpose in covering this topic is to give the
reader the tools they need to generate the types of random variables that
often arise in practice and to provide examples illustrating the methods. We
first discuss general techniques for generating random variables, such as the
inverse transformation and acceptance-rejection methods. We then provide
algorithms and MATLAB code for generating random variables for some
useful distributions.

4.2 General Techniques for Generating Random Variables

m
Numbe
for
m
Numbe
Numbe
Random
Random
ni
U
U
Un
U n i ii for for for m m Random Random r Numbe rs r r s s s
n
Most methods for generating random variables start with random numbers
,
(
that are uniformly distributed on the interval 01) . We will denote these
random variables by the letter U. With the advent of computers, we now have
© 2002 by Chapman & Hall/CRC