Let $Y$ be a random variable with probability density function $f_Y(y)$ Then the hazard function $h(y)$ is defined to be: $$h(y) = \frac{f_Y(y)}{1 - F_Y(y)} = \frac{f_Y(y)}{S(y)},$$ where $S(y)$ is the survivor function and $Y$ is the survival time.

The hazard function is the rate of probability of death (non survival) is changing at time $Y=y$ given survival up to time $y$ $$h(y) = \lim_{\Delta y\rightarrow 0} \frac {P(y\leq Y \leq y+\Delta y \mid Y > y)}{\Delta y}.$$

The cumulative hazard function, $H(y)$ of $Y$ is defined as $$H(y) = \int_{-\infty}^{y} h(t) dt.$$

From this definition, we see that $H(y)=-\operatorname{ln}S(y)$

Examples. The hazard functions for the three most widely used probability density functions for survival time are:

• The exponential distribution, with $h(y)=\gamma$
• The Weibull distribution, with $h(y)=\gamma y^{\gamma-1}$ using the standard Weibull distribution.
• The extreme-value distribution, with $h(y)=\frac{1}{\beta}\operatorname{exp}(\frac{y-\alpha}{\beta})$