Truncated normal distribution ($T{N}_A(\mu ,{\sigma }^2)$)

A truncated normal distribution is a normal distribution conditioned to lie in a specified subset of the real line. Its density is the original Gaussian density divided by the probability of the retained set.

To draw $N(\mu ,1)$ restricted to a positive value, let $a=\Phi (-\mu )$ and return $\mu +{\Phi }^{-1}(a+(1-a)U)$ for uniform $U\in (0,1)$. For restriction to a negative value use $\mu +{\Phi }^{-1}(aU)$. These are inverse transform sampling constructions. Repeatedly drawing an untruncated normal until its sign agrees is also exact, but inefficient for rare signs.