RE: Generating a normal 0,1

• To: mathgroup at yoda.ncsa.uiuc.edu
• Subject: RE: Generating a normal 0,1
• From: blachman%gtewd.dnet at gte.com (NELSON M. BLACHMAN)
• Date: Wed, 23 Jan 91 17:11:49 -0500

```  Generating a single standard normal random variable x is somewhat hard and
slow, but generating two independent ones, x and y, is easy.

Let u and v be two independent reals distributed uniformly between 0 and 1.
Then
x = Sqrt[-2 Log[u]] Cos[2 Pi v]
and
y = Sqrt[-2 Log[u]] Sin[2 Pi v]

are independent zero-mean normal random variables with unit variance.

This is established by noticing that the angle ArcTan[y/x] must have a
uniform distribution like v/(2 Pi), and r = Sqrt[x^2 + y^2] must have a
Rayleigh distribution with cumulative distribution function  1 - E^(-r^2/2),
which can be set equal to 1 - u.

Nelson M. Blachman
GTE Government Systems Corp.
Mountain View, California
blachman%gtewd.dnet at gte.com

```

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