Acceptance-Rejection Sampling

This method generates random vectors ${\bf y}_i$ with normally distributed random components, and estimates $P({\bf b})$ using

$$\bar{P} = \frac{1}{N}\sum_{j=1}^N R(C{\bf y}_j)),$$

where

$$R({\bf x}) = \left\{ \begin{array}{cl} 1 & x_i \leq b_i \mbo... ... 1 \leq i \leq m \\ 0 & \mbox{otherwise} \end{array}. \right.$$

Deák explains that in this case a standard error estimate is given by $E = \sqrt{\bar{P}(1-\bar{P})/N}$. Because $E < 0.5/\sqrt{N}$, at least one decimal digit of absolute accuracy is expected with probability 0.95 when N = 100.