Integrated tail distribution ($f_I(x)=\mathbb{P}(X>x)/\mathbb{E}X$)

For a positive random variable $X$ with finite nonzero expected value $\mu$, its integrated tail law has density $f_I(x)=\mathbb{P}(X>x)/\mu$ on $x\ge 0$. The tail integral formula for moments normalizes it. For $r>0$ its exponential transform is $\int e^{rx}{f}_I(x)dx=[M_X(r)-1]/(\mu r)$, with extended-value interpretation when needed.