## Encyclopedia > Box-Muller transform

Article Content

# 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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Montenegro ... Podgorica (Titograd up to 1992), with 130,000 inhabitants, Niksic[?] (60,000) and Pljevlja[?] (22,000). History of Montenegro List of cities in Montenegro Geography ...