## Standard normal distribution

Probability density function of the standard normal distribution with mean 0 and standard deviation 1, is as follows,
$f(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}$

(1)

Box-Muller method converts $U_1,U_2$ from standard uniform random number uniformly distributed interval $(0,1]$ by the following equation, it is possible to obtain $Z_1, Z_2$ mutually independent standard normal random numbers.

$Z_1=\sqrt{-2\log _e U_1}\cos \left(2 \pi U_2\right) $

(2)

$Z_2=\sqrt{-2\log _e U_2}\cos \left(2 \pi U_1\right)$

In addition, there are method of using the central limit theorem by Yoshikazu Shimizu, rejection method and so on.

Reference:

- JIS Z 9031:2012, Procedure for random number generation and randomization

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