Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 38 / 2 / b

On $\{T>c\}$, $X=c$ and the integrated hazard equals the constant $H(c)$. ThusFor the following finite calculations take $F(c)>0$; endpoint cases follow by a limit.

The event contribution is an ordinary density integral. Since $f=-F^{\prime }$ and $H^{\prime }=h$, integration by parts givesConsequently

Adding the event and censored contributions cancels the boundary terms:This is the mean accumulated hazard before fixed censoring identity. If $F(c)=0$, interpret the boundary term by its limiting value: $FH=-F\log F\to 0$. The identity then gives one.