If $T$ is future time to dementia from age 50, its survival function is
$$
\mathbb P(T>t)=
\begin{cases}
e^{-t/10},&0\leq t\leq10,\\
e^{-1}e^{-(t-10)/5},&t>10.
\end{cases}
$$
The <tail-sum formula for expectation> gives the expected future time alive without dementia:
$$
\mathbb ET
=\int_0^{10}e^{-t/10}\,dt
+\int_{10}^{\infty}e^{-1}e^{-(t-10)/5}\,dt
=10(1-e^{-1})+5e^{-1}
=\boxed{10-5e^{-1}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-207/1/b/ii/solution