Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 1 / b / ii / Solution

For individual $i$, let $X_i=\min (T_i,c_i)$ and ${\Delta }_i=1_{\{T_i\le c_i\}}$. The preceding expectation calculation applies separately at each fixed censoring time:Summing and using linearity of expectation, with $D={\sum }_i{\Delta }_i$ the failure count, givesThis identity needs the stated marginal exponential distributions, but not independence between individuals; independence will be needed for the product likelihood.