Page 103 - Computational Statistics Handbook with MATLAB

90 Computational Statistics Handbook with MATLAB

Fx() = 1 – e – λx ; 0 < x < ∞ . (4.9)

Letting

u = F x() = 1 – e – λx , (4.10)

we can solve for x, as follows

u = 1 – e – λx
e – λx = 1 – u
– λx = log ( 1 – u)
1
u
x = – ---log ( 1 – ).
λ
By making note of the fact that 1 – u is also uniformly distributed over the
interval (0,1), we can generate exponential random variables with parameter
λ using the transformation

1
X = – ---log () . (4.11)
U
λ
Example 4.6
The following MATLAB code will generate exponential random variables for
λ
a given .
% Set up the parameters.
lam = 2;
n = 1000;
% Generate the random variables.
uni = rand(1,n);
X = -log(uni)/lam;
We can generate a set of random variables and plot them to verify that the
function does yield exponentially distributed random variables. We plot a
histogram of the results along with the theoretical probability density func-
tion in Figure 4.4. The MATLAB code given below shows how we did this.

% Get the values to draw the theoretical curve.
x = 0:.1:5;
% This is a function in the Statistics Toolbox.
y = exppdf(x,1/2);
% Get the information for the histogram.
[N,h] = hist(X,10);
% Change bar heights to make it correspond to

© 2002 by Chapman & Hall/CRC