Parametric Survival Distributions Cheatsheet

Cheatsheet: Parametric Survival Distributions

Comparison Table

Distribution	$S(t)$	$h(t)$	$f(t)=S(t)\cdot h(t)$	$H(t)$	$E(T)$	$V(T)$
Exp	$e^{-\lambda t}$	$\lambda$	$\lambda e^{-\lambda t}$	$\lambda t$	$\frac{1}{\lambda }$	$\frac{1}{{\lambda }^2}$
Weibull	$e^{-\lambda t^{\alpha }}$	$\alpha \lambda t^{\alpha -1}$	$\alpha \lambda t^{\alpha -1}{e}^{-\lambda t^{\alpha }}$	$\lambda t^{\alpha }$	—	—
Gamma	—	—	$\frac{{\lambda }^{\alpha }{t}^{\alpha -1}{e}^{-\lambda t}}{\Gamma (\alpha )}$	—	$\frac{\alpha }{\lambda }$	$\frac{\alpha }{{\lambda }^2}$

Shape Parameter (Weibull)

• $\alpha <1$: Decreasing hazard
• $\alpha =1$: Constant hazard (exponential)
• $\alpha >1$: Increasing hazard