Marsaglia polar method

The Marsaglia polar method^[1] is a pseudo-random number sampling method for generating a pair of independent standard normal random variables.^[2]

Standard normal random variables are frequently used in computer science, computational statistics, and in particular, in applications of the Monte Carlo method.

The polar method works by choosing random points (x, y) in the square −1 < x < 1, −1 < y < 1 until

${\displaystyle 0<s=x^{2}+y^{2}<1,\,}$

and then returning the required pair of normal random variables as

${\displaystyle x{\sqrt {\frac {-2\ln(s)}{s}}}\,,\ \ y{\sqrt {\frac {-2\ln(s)}{s}}},}$

or, equivalently,

${\displaystyle {\frac {x}{\sqrt {s}}}{\sqrt {-2\ln(s)}}\,,\ \ {\frac {y}{\sqrt {s}}}{\sqrt {-2\ln(s)}},}$

where ${\displaystyle x/{\sqrt {s}}}$ and ${\displaystyle y/{\sqrt {s}}}$ represent the cosine and sine of the angle that the vector (x, y) makes with x axis.