 Solution

Use the ratio-of-uniforms method onWriting $x=v/u$ gives $u\le e^{-x^2/(4{\sigma }^2)}$ and $v=ux$. Consequentlybecause the derivative vanishes at $x=\sigma \sqrt{2}$. Thus the normal ratio-of-uniforms envelope is the rectangle $0<u<1$, $|v|<\sigma \sqrt{2/e}$.

Generate independent $U_1,U_2$ with uniform distribution on $(0,1)$, set $u=U_1$ and $v=(2U_2-1)\sigma \sqrt{2/e}$, and accept precisely whenReject and repeat otherwise. On acceptance return $X=v/u$. One can compare logarithms instead of exponentials to avoid underflow.

For a proof, the change of variables $(u,x)\mapsto (u,v=ux)$ has Jacobian determinant $u$. A uniform point in $\mathcal{R}$ therefore gives an $x$-density proportional toNormalizing yields the required normal distribution with mean zero and variance ${\sigma }^2$. The region area is $\frac{1}{2}\int h(x)dx=\sigma \sqrt{\pi /2}$; dividing it by the rectangle area $2\sigma \sqrt{2/e}$ gives acceptance probability $\sqrt{\pi e}/4$.