Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 2 / a

For a continuous nonnegative event time with hazard function $h$, the cumulative hazard function isIts derivative is the instantaneous hazard wherever the derivative exists.

Writing $F$ for the survival function here, the relation $F^{\prime }(t)=-h(t)F(t)$ with $F(0)=1$ givesThis notation uses $F$ for survival rather than for the cumulative distribution function.

The continuous failure-time probability density function is the negative derivative of the survival function:The minus sign is essential because survivor probability decreases with time.