<P> Then Z and Z are independent random variables with a standard normal distribution . </P> <P> The derivation is based on a property of a two - dimensional Cartesian system, where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for R and Θ (shown above) in the corresponding polar coordinates are also independent and can be expressed as </P> <Dl> <Dd> R 2 = − 2 ⋅ ln ⁡ U 1 (\ displaystyle R ^ (2) = - 2 \ cdot \ ln U_ (1) \,) </Dd> </Dl> <Dd> R 2 = − 2 ⋅ ln ⁡ U 1 (\ displaystyle R ^ (2) = - 2 \ cdot \ ln U_ (1) \,) </Dd>

A note on the generation of random normal deviates