HTML version of Scripted Foils prepared 12 November 1996

Foil 3 Generation of Gaussian Distributions

From CPS615-End of Basic Overview of Random Numbers and First Part of Monte Carlo Integration Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 22 October 96. by Geoffrey C. Fox *
Secs 83.5
1 "A Small Miracle" asserts that:
2 If x1 x2 are uniformly distributed in [0,1] -- Then:
  • g1 = (-2lnx1)1/2 cos 2px2
  • g2 = (-2lnx1)1/2 sin 2px2
3 are Gaussianly distributed
4 with mean = 0 and standard deviation = 1
5 while g1 and g2 are independent.
6 Proof: Consider
7 Integral:
8 with g1 g2 going to Polar coordinates (r=radius, angle)
9 and then transform to x1and x2 by
10 (-2lnx1)1/2 = radius i.e. x1=exp(-r2/2) and
11 2px2 = angle
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