Mean Life

Definition

Let $T$ be Survival Time

Mean life (also called expected life time or mean lifetime) is the expected value of the survival time:

$\mu =E[T]$

Interpretation

The average survival time across all subjects. Also known as usia harapan hidup (life expectancy).

Formula

Mean life can be computed from the survival function:

$\mu =E[T]={\int }_0^{\infty }S(t)dt$

Relationship with Mean Residual Life

Mean life is a special case of Mean Residual Life at $t=0$:

$\mu =E[T]=E[T-0\mid T>0]=\text{mrl}(0)$

Comparison with Median Life

Aspect	Mean Life $\mu$	Median Life $t_{0.5}$
Definition	Average: $E[T]$	50th percentile: $S(t)=0.5$
Formula	${\int }_0^{\infty }S(t)dt$	Solve $S(t)=0.5$ for $t$
Interpretation	Expected lifetime	Half fail before, half after
Affected by	All values (sensitive to outliers)	Only the middle value (robust)

Example

Exponential distribution with $S(t)=e^{-\lambda t}$:

• Mean life: $\mu ={\int }_0^{\infty }{e}^{-\lambda t}dt=\frac{1}{\lambda }$
• Median life: $e^{-\lambda t_{0.5}}=0.5\Rightarrow t_{0.5}=\frac{\ln 2}{\lambda }\approx \frac{0.693}{\lambda }$

Note: For exponential, mean > median (right-skewed distribution).

Related

• Median Life
• Mean Residual Life
• Survival Function
• Survival Quantities Cheatsheet