Integrated cumulative distribution function on a bounded interval (${\int }_l^r{F}_X(t)dt=r-\mathbb{E}[X]$)

If a random variable $X$ takes values in $[l,r]$, its cumulative distribution function $F_X$ satisfiesIndeed ${\int }_l^r{1}_{\{X\le t\}}dt=r-X$ 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 ${\int }_l^r{\int }_l^t{f}_X(x)dxdt={\int }_l^r(r-x)f_X(x)dx$. 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.