In the previous section, a simple, but less accurate method of generating a normal distribution was presented. Here we consider a direct transormation.

- Let be two standard normal random variates.
- Plot the two as a point in the plane and represent them in a
polar coordinate system as

- It is known that
has the chi-square
distribution with 2 degrees of freedom, which is equivalent to an
exponential distribution with mean 2

thus the raidus*B*can be generated using (9.3)

- So a normal distribution can be generated by any one of the
following.

where and are uniformly distributed over (0,1). - To obtain normail variates with mean and variance
, transform