Survival Quantities Cheatsheet

Cheatsheet: Survival Quantities

Basic Formulas

$$\begin{array}{rl}S(t)& =1-F(t)\\ h(t)& =\frac{f(t)}{S(t)}\\ H(t)& =\int \_0^{t}h(u)du=-\ln S(t)\\ mrl(t)& =E\text{[}T-t\mid T>t]=\frac{\int \_t^{\infty }S(u)du}{S(t)}\\ \mu & =E\text{[}T]=\int \_0^{\infty }S(u)du\end{array}$$

Relationships

Quantity	Formula	Derivation
$S(t)$	$\exp \text{[}-H(t)]$	From$H(t)=-\ln S(t)$
$h(t)$	$\frac{f(t)}{S(t)}$	Proof
$H(t)$	$-\ln S(t)$	Integral of $h(t)$

Total Time on Test (TTT)

Cumulative operating time until last failure.