Page 101 - Computational Statistics Handbook with MATLAB

88 Computational Statistics Handbook with MATLAB

PROCEDURE - REJECTION METHOD (DISCRETE)

that is easy to sample from.
1. Choose a probability mass function q i
2. Find a constant c such that p Y < cq Y .
.
3. Generate a random number Y from the density q i
4. Generate a uniform random number U.
5. If

U ≤ -------- ,
p Y
cq Y
then deliver X = Y , else go to step 3.

Example 4.5
In this example, we use the discrete form of the acceptance-rejection method
to generate random variables according to the probability mass function
defined as follows
(
PX = 1) = 0.15,
(
PX = 2) = 0.22,
(
PX = 3) = 0.33,
(
PX = 4) = 0.10,
(
PX = 5) = 0.20.
be the discrete uniform distribution on 1 … 5, ,
We let q Y , where the proba-
bility mass function is given by

1
,
-
q y = --; y = 1 …, . 5
5
We describe a method for generating random variables from the discrete uni-
form distribution in a later section. The value for c is obtained as the maxi-
⁄
mum value of p y q y , which is 1.65. This quantity is obtained by taking the
, which is PX =(
maximum p y 3) = 0.33 , and dividing by 1/5:
max p y )
(
--------------------- = 0.33 × 5 = 1.65 .
⁄
15
The steps for generating the variates are:

© 2002 by Chapman & Hall/CRC