Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 4 / b

For $s,\tau \ge 0$, use conditional probability and the survival function:It is independent of $s$ and equals $P(T>\tau )$. This is the memoryless property of the exponential distribution, equivalent here to a constant hazard function $\alpha$.

Assume $C,Q>0$. The expected net profit per item is $\Pi (\alpha )=Q\mathbb{E}[T]-C/{\alpha }^2=Q/\alpha -C/{\alpha }^2$. Put $u=1/\alpha >0$ and complete the square:This quadratic function has a unique global maximum at $u=Q/(2C)$. HenceThe calculation uses the expected operating time, rather than replacing lifetime by an arbitrary fixed time.