Page 98 - Computational Statistics Handbook with MATLAB

Chapter 4: Generating Random Variables 85

else
X(i) = x(3); % It has to be between 0.5 and 1.
end
end
One way to verify that our random variables are from the desired distribu-
tion is to look at the relative frequency of each x.

% Find the proportion of each number.
x0 = length(find(X==0))/100;
x1 = length(find(X==1))/100;
x2 = length(find(X==2))/100;
The resulting estimated probabilities are

ˆ
P x =( x 0 ) = 0.26
ˆ
P x =( x 1 ) = 0.21
ˆ
P x =( x 2 ) = 0.53.

These values are reasonable when compared with the desired probability
mass values.

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In some cases, we might have a simple method for generating a random vari-
able from one density, say gy() , instead of the density we are seeking. We can
use this density to generate from the desired continuous density f x() . We
first generate a random number Y from gy() and accept the value with a
probability proportional to the ratio f Y() (⁄ gY()) .
If we define c as a constant that satisfies
fy()
----------- ≤ c; for all y , (4.6)
gy()
then we can generate the desired variates using the procedure outlined
below. The constant c is needed because we might have to adjust the height
of gy() to ensure that it is above f y() . We generate points from cg y() , and
those points that are inside the curve f y() are accepted as belonging to the
desired density. Those that are outside are rejected. It is best to keep the num-
ber of rejected variates small for maximum efficiency.
© 2002 by Chapman & Hall/CRC