Solution
= Solution
The <expectation> follows by <integration by parts>, with $u=t$ and $dv=e^{-t/25}\,dt/25$:
$$
\mathbb E[T]=\int_0^\infty t\,\frac1{25}e^{-t/25}\,dt
=\left[-te^{-t/25}\right]_0^\infty+\int_0^\infty e^{-t/25}\,dt.
$$
The endpoint term vanishes because the <exponential function> decays faster than $t$ grows. Hence the mean lifetime is
$$
\boxed{\mathbb E[T]=25\ \text{months}.}
$$