Mean residual life ($m_X(t)=\mathbb{E}[X-t\mid X>t]$)

For a nonnegative random variable with finite expected value and $\mathbb{P}(X>t)>0$, its mean residual life is the remaining conditional expected value after surviving past $t$. The tail integral formula for moments gives $m_X(t)={\int }_t^{\infty }\mathbb{P}(X>x)dx/\mathbb{P}(X>t)$. The exponential distribution has constant mean residual life, equal to its original mean. In excess of loss reinsurance, the total variance stationary condition for excess of loss sets a candidate retention equal to this quantity.