Page 96 - Computational Statistics Handbook with MATLAB

Chapter 4: Generating Random Variables 83

We illustrate this procedure using a simple example.

Example 4.2
We would like to simulate a discrete random variable X that has probability
mass function given by
(
PX = 0) = 0.3,
(
PX = 1) = 0.2,
(
PX = 2) = 0.5.

The cumulative distribution function is

 0; x < 0

 0.3; 0 ≤ x < 1
Fx() = 
 0.5; 1 ≤ x < 2
 2 ≤
 1.0; x.

We generate random variables for X according to the following scheme

 0; U ≤ 0.3

X =  1; 0.3 < U ≤ 0.5

 2; 0.5 < U ≤ 1.

This is easily implemented in MATLAB and is left as an exercise. The proce-
dure is illustrated in Figure 4.2, for the situation where a uniform random
variable 0.73 was generated. Note that this would return the variate x = 2 .

We now outline the algorithmic technique for this procedure. This will be
useful when we describe a method for generating Poisson random variables.

PROCEDURE - INVERSE TRANSFORM (DISCRETE)
,
,
1. Define a probability mass function for x i i = 1 … k . Note that k
,
could grow infinitely.
2. Generate a uniform random number U.
3. If U ≤ p 0 deliver X = x 0
4. else if U ≤ p 0 + p 1 deliver X = x 1
5. else if U ≤ p 0 + p 1 + p 2 deliver X = x 2

© 2002 by Chapman & Hall/CRC