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There are two primary methods for generating normally distributed
random variables. The first method relies on the central limit theorem
which states that if E and Var
, then for
independent and identically distributed samples from any distribution
has an approximate
distribution, where is the sample size. The utility in this first method is that it
is very easy to remember off the top of your head and is relatively
easy to compute with a computer.

The second method uses a direct transformation, and, while being just
as easy to compute using a computer, is a little tricky to
remember. This method is called the Box-Muller algorithm. The steps
involved are:

- Generate to independent uniform(0,1) variables, , .
- Let
and
- Let
and
.

where and are independent normal(0,1) random variables. See
Appendix A.2 for an Octave program that implements
this algorithm.

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Frank Starmer
2004-05-19