Editing Box–Muller transform (section)

==Basic form==
Suppose {{math|''U''<sub>1</sub>}} and {{math|''U''<sub>2</sub>}} are independent samples chosen from the uniform distribution on the [[Interval (mathematics)|unit interval]] {{open-open|0, 1}}. Let
<math display="block">Z_0 = R \cos(\Theta) =\sqrt{-2 \ln U_1} \cos(2 \pi U_2)\,</math>
and
<math display="block">Z_1 = R \sin(\Theta) = \sqrt{-2 \ln U_1} \sin(2 \pi U_2).\,</math>

Then ''Z''<sub>0</sub> and ''Z''<sub>1</sub> are [[statistical independence|independent]] random variables with a [[standard normal distribution]].

The derivation<ref>Sheldon Ross, ''A First Course in Probability'', (2002), pp. 279–281</ref> is based on a property of a two-dimensional [[Cartesian_coordinate_system|Cartesian system]], where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for {{math|''R''<sup>2</sup>}} and {{mvar|Θ}} (shown above) in the corresponding polar coordinates are also independent and can be expressed as
<math display="block">R^2 = -2\cdot\ln U_1\,</math>
and
<math display="block">\Theta = 2\pi U_2. \,</math>

Because {{math|''R''<sup>2</sup>}} is the square of the norm of the standard [[bivariate normal]] variable {{math|(''X'', ''Y'')}}, it has the [[chi-squared distribution]] with two degrees of freedom. In the special case of two degrees of freedom, the chi-squared distribution coincides with the [[exponential distribution]], and the equation for {{math|''R''<sup>2</sup>}} above is a simple way of generating the required exponential variate.