Solution
= Solution
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:
$$
\mathbb ET
=\int_0^{10}e^{-t/10}\,dt
+\int_{10}^{\infty}e^{-1}e^{-(t-10)/5}\,dt
=10(1-e^{-1})+5e^{-1}
=\boxed{10-5e^{-1}}.
$$