Hazard function

\[\begin{equation} \begin{aligned} \lambda(t) & = \lim_{\Delta t \to 0} \frac{P\big[t \leq T < t + \Delta t | T \geq t \big]}{\Delta t} \\\\ & = \lim_{\Delta t \to 0} \frac{P\big[t \leq T < t + \Delta t \big] / P\big[T \geq t \big]}{\Delta t} \\\\ & = \lim_{\Delta t \to 0} \frac{\big(F(t+\Delta t)-F(t)\big) / \Delta t}{S(t)} \\\\ & = \frac{\text{d} F(t) / \text{d} t}{S(t)} \\\\ & = \frac{f(t)}{S(t)} \\\\ & = - \frac{\text{d}S(t) / \text{d} t}{S(t)} \\\\ \lambda(t) & = \frac{-\text{d} \ln \big(S(t)\big)}{\text{d} t} \end{aligned} \tag{6.3} \end{equation}\]