Mean Residual Life Formula

Theorem

Let $mrl(t)=E[T-t\mid T>t]$. Then $mrl(t)=\frac{{\int }_t^{\infty }S(u)du}{S(t)}$

Proof

By definition of Conditional Probability. For aContinuous Random Variable, the conditional density given $T>t$ is:

$f(u\mid T>t)=\frac{f(u)}{\mathrm{Pr}(T>t)}=\frac{f(u)}{S(t)},u>t$

Let $T=u$, then $T-t=u-t$. So, by definition of Expectation for continuous random variable:

$E[T-t\mid T>t]={\int }_t^{\infty }\underset{\text{value of }T-t}{\underbrace{(u-t)}}\cdot \underset{\text{pdf of }T}{\underbrace{\frac{f(u)}{S(t)}}}du$Integration by parts with $v^{\prime }=f(u)$, $w=u-t$:

${\int }_t^{\infty }(u-t)f(u)du={\left[-(u-t)S(u)\right]}_t^{\infty }+{\int }_t^{\infty }S(u)du$

The boundary term: at $u=t$, $(t-t)S(t)=0$. At $u=\infty$, $(u-t)S(u)\to 0$ (assuming finite mean). So the boundary term vanishes, leaving:

${\int }_t^{\infty }(u-t)f(u)du={\int }_t^{\infty }S(u)du$

Therefore:

$\boxed{mrl(t)=\frac{{\int }_t^{\infty }S(u)du}{S(t)}}$

Related

• Mean Residual Life
• Survival Function