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

For $t_0\ge 0$, integration of the exponential distribution gives its cumulative distribution function:For $t_0<0$ the event is empty and its probability is zero.

The probability density function is normalized because ${\int }_0^{\infty }\alpha e^{-\alpha t}dt=1$. Integration by parts gives $\mathbb{E}[T]={\int }_0^{\infty }t\alpha e^{-\alpha t}dt=1/\alpha$ and $\mathbb{E}[T^2]=2/{\alpha }^2$. ThereforeThese are the mean and variance of the exponential distribution with rate $\alpha$.

The survival function of this exponential distribution is $P(T>t)=e^{-\alpha t}$. At the expected value found above,

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.