= Truncated distribution
{wiki}
A <truncated distribution> is the <conditional distribution>[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 is
$$
\mu_A(B)=\mathbb P(X\in B\mid X\in A)=\frac{\mu(B\cap A)}{\mu(A)},\qquad \mu(A)>0.
$$
The denominator renormalizes the retained <probability mass> to one. If $X$ has a <probability density function> $f$, the truncated <probability density function>[density] is $f_A(x)=f(x)\mathbf1_A(x)/\mu(A)$; the same restriction and <normalization of a probability distribution>[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 <probability density function>[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.
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