Hazard Function Derivation

Theorem

Let:

• $f(t)$ = probability density function
• $S(t)$ = survival function

Then $h(t)=\frac{f(t)}{S(t)}$

Proof

$$\begin{array}{rl}h(t)& =\underset{\Delta t\to 0}{\lim }\frac{1}{\Delta t}\cdot \frac{\mathrm{Pr}(t<T\le t+\Delta t)}{\mathrm{Pr}(T>t)}\\ & =\underset{\Delta t\to 0}{\lim }\frac{F(t+\Delta t)-F(t)}{\Delta t\cdot S(t)}\end{array}$$

The limit ${\lim }_{\Delta t\to 0}\frac{F(t+\Delta t)-F(t)}{\Delta t}$ is by definition $F^{\prime }(t)=f(t)$.

Therefore: $\boxed{h(t)=\frac{f(t)}{S(t)}}$

Related

• Hazard Function
• Survival Function