Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 7

The probability density function is nonnegative and vanishes on negative lifetimes. Since $0.04=1/25$, its normalization of a probability distribution isIt is an exponential distribution with rate $1/25$ per month.

Integrating the probability density function over the stated interval givesThe strict endpoint does not affect the probability for this continuous probability distribution.

The survival function of this exponential distribution is $\mathbb{P}(T>t)=e^{-t/25}$ for $t\ge 0$. Therefore

Subtract the two values of the survival function, or integrate the probability density function directly:The final zero records the requested third significant figure.

The expectation follows by integration by parts, with $u=t$ and $dv=e^{-t/25}dt/25$:The endpoint term vanishes because the exponential function decays faster than $t$ grows. Hence the mean lifetime is