= Integrated cumulative distribution function on a bounded interval
{title2=$\int_l^rF_X(t)\,dt=r-\mathbb E[X]$}
If a <random variable> $X$ takes values in $[l,r]$, its <cumulative distribution function> $F_X$ satisfies
$$
\boxed{\int_l^r F_X(t)\,dt=r-\mathbb E[X].}
$$
Indeed $\int_l^r\mathbf1_{\{X\leq t\}}\,dt=r-X$ for every outcome. Taking <expected values> and using <Tonelli's theorem> proves the formula, including when the <probability distribution> has atoms. For a <probability density function> $f_X$, the same proof reverses the order in $\int_l^r\int_l^t f_X(x)\,dx\,dt=\int_l^r(r-x)f_X(x)\,dx$. Thus $F_X(t)/(r-\mathbb E[X])$, restricted to $[l,r]$, is a <probability density function> whenever $\mathbb E[X]<r$. The reflected construction using $1-F_X(t)$ has normalising constant $\mathbb E[X]-l$ and is closely related to an <integrated tail distribution>.
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