Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 / 4 / b

One method applies inverse transform sampling to the standard normal cumulative distribution function: $Z_i={\Phi }^{-1}(U_i)$, followed by $X_i=\mu +\sigma Z_i$ for mean $\mu$ and standard deviation $\sigma >0$. Numerical approximations to ${\Phi }^{-1}$ are needed, and the endpoints $U=0,1$ must be avoided.

A convenient alternative is the Box-Muller transform. From independent uniforms $U_1,U_2\in (0,1)$ setThe radial probability density function is $r{e}^{-r^2/2}$ for $r>0$, and the angle is independently uniform on $[0,2\pi )$. Dividing their joint density by the polar-coordinate Jacobian determinant $r$ givesThus the outputs are independent standard normal distribution draws. Apply $\mu +\sigma Z_j$ to obtain other normal means and variances. With finite pseudorandom number generators, this is an approximation to the ideal independent-uniform model; poor dependence in the input stream is not cured by the transform.