## Encyclopedia > Box-Muller transform

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# Box-Muller transform

A Box-Muller transform is a method of generating pairs of independent normally distributed random numbers, given a source of uniformly distributed random numbers. There are two kinds:

1. Given r and φ independently uniformly distributed in (0,1], compute:
$z_0 = \cos(2 \pi \varphi) \cdot \sqrt{-2 \ln r}$
and
$z_1 = \sin(2 \pi \varphi) \cdot \sqrt{-2 \ln r}$
2. Given x and y independently uniformly distributed in [-1,1], set r = x2+y2. If r = 0 or r > 1, throw them away and try another pair (x, y). Thus, all points left after this filtering process will be uniformly distributed within a unit circle. Then, for these filtered points, compute:
$z_0 = x \cdot \sqrt{\frac{-2 \ln r}{r}}$
and
$z_1 = y \cdot \sqrt{\frac{-2 \ln r}{r}}$

The second method is faster because it uses only one transcendental function[?] instead of three, even though it throws away 21% of the numbers.

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