Separate HTML for Basic Foil 10 Generation of Gaussian Distributions

Basic HTML version of Foils prepared 21 October 1995

Foil 10 Generation of Gaussian Distributions

From CPS615 Foils -- set D:Statistics and Random Numbers CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. by Geoffrey C. Fox

"A Small Miracle" asserts that:

If x1 x2 are uniformly distributed in [0,1] -- Then:

g1 = (-2lnx1)1/2 cos 2px2
g2 = (-2lnx1)1/2 sin 2px2

are Gaussianly distributed

with mean = 0 and standard deviation = 1

while g1 and g2 are independent.

Proof: Consider

Integral:

with g1 g2 going to Polar coordinates (r=radius, angle)

and then transform to x1and x2 by

(-2lnx1)1/2 = radius i.e. x1=exp(-r2/2) and

2px2 = angle

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