 Truncated distribution

A truncated distribution is the conditional probability distribution obtained by restricting a random variable $X$ to a specified event $X\in A$ with positive probability. If its original law is $\mu$, the truncated law isThe denominator renormalizes the retained probability mass to one. If $X$ has a probability density function $f$, the truncated density is $f_A(x)=f(x)1_A(x)/\mu (A)$; the same restriction and normalization apply to a probability mass function.

For example, truncate a uniform distribution on $(0,10)$ to values at least six. The retained event has probability $2/5$, so the new density is $1/4$ on $(6,10)$, and its mean is eight. This is the law obtained by repeatedly sampling from the original law and rejecting values outside the retained set; rejection sampling becomes inefficient when the retained event is rare.

A zero-probability restriction cannot be normalized by the displayed formula. Conditioning a continuous random variable to equal a specified value requires an additional conditional-law construction or limiting convention, rather than division by zero.