Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 4 / b

One route is inverse transform sampling: if $\Phi$ is the standard normal cumulative distribution function, then $\mu +\sigma {\Phi }^{-1}(U)$ has the normal distribution with mean $\mu$ and variance ${\sigma }^2$.

Alternatively the Box-Muller transform uses two independent uniform inputs $U_1,U_2\in (0,1)$:To verify this, set $R=\sqrt{-2\log U_1}$ and $\Theta =2\pi U_2$. The radial density is $r{e}^{-r^2/2}$ for $r>0$, and the angle is independent and uniform. Their joint density is $r{e}^{-r^2/2}/(2\pi )$. Changing to Cartesian coordinates divides by the Jacobian determinant $r$, givingThis factors into two standard normal densities, proving both normality and independence. Then $\mu +\sigma Z_j$ provides the required means and variances. Implementations must avoid the zero input to the logarithm; ideal continuous uniforms hit it with probability zero.