Page 108 - Computational Statistics Handbook with MATLAB

Chapter 4: Generating Random Variables 95

The chi-square distribution is useful in situations where we need to system-
atically investigate the behavior of a statistic by changing the skewness of the
distribution. As the degrees of freedom for a chi-square increases, the distri-
bution changes from being right skewed to one approaching normality and
symmetry.

Bet
aa
BBeett taa
Be
The beta distribution is useful in simulations because it covers a wide range
α
of distribution shapes, depending on the values of the parameters and β.
These shapes include skewed, uniform, approximately normal, and a bimo-
dal distribution with an interior dip.
First, we describe a simple approach for generating beta random variables
β
α
with parameters and , when both are integers [Rubinstein, 1981; Gentle,
k
1998]. It is known [David, 1981] that the -th order statistic of n uniform (0,1)
variates is distributed according to a beta distribution with parameters and
k
k
n – + 1 . This means that we can generate random variables from the beta
distribution using the following procedure.
PROCEDURE - BETA RANDOM VARIABLES (INTEGER PARAMETERS)
,
,
1. Generate α + β – 1 uniform random numbers: U 1 … U α + β – 1
2. Deliver X = U α() which is the α -th order statistic.

One simple way to generate random variates from the beta distribution is
are indepen-
to use the following result from Rubinstein [1981]. If Y 1 and Y 2
has a gamma distribution with parameters
dent random variables, where Y 1
β
α and 1, and Y 2 follows a gamma distribution with parameters and 1, then
Y 1
X = ------------------ (4.17)
Y + Y 2
1
α
β
is from a beta distribution with parameters and . This is the method that
is used in the MATLAB Statistics Toolbox function betarnd that generates
random variates from the beta distribution. We illustrate the use of betarnd
in the following example.

Example 4.9
We use this example to illustrate the use of the MATLAB Statistics Toolbox
function that generates beta random variables. In general, most of these tool-
box functions for generating random variables use the following general syn-
tax:
rvs = pdfrnd(par1,par2,nrow,ncol);

© 2002 by Chapman & Hall/CRC