Could someone please provide some pointers on how to complete the following question?
Consider a Yule process $\{X(t): t \geq 0\}$. Suppose that $X(0) = 1$ and the process stops at time $T$ and is replaced by an immigration process where departures occur according to a Poisson process with rate $\mu$. Let $\tau$ be the time since $T$ for the population to vanish. Find the pdf of $\tau$ and show that $E(\tau) = e^{\lambda T}/\mu$.
Using the Kolmogorov backward equations, it can be shown by induction that $$ \mathbb P(X(T) = n\mid X(0)= 1) = e^{-\lambda T}(1-e^{-\lambda T})^{n-1},\ n\geqslant 1. $$ (See here for a proof.)
In other words, $X(T)$ has geometric distribution with parameter $e^{-\lambda T}$. Then $\tau = \sum_{j=1}^N S_j$ where $N\sim\mathrm{Geo}(e^{-\lambda T})$ and $S_j\stackrel{\mathrm{i.i.d.}}\sim \mathrm{Expo}(\mu)$. A simple computation shows that $\tau$ has exponential distribution with parameter $e^{-\lambda T}\mu$. Hence the density of $\tau$ is $$ e^{-\lambda T}\mu e^{-e^{-\lambda T}\mu t}\mathsf 1_{(0,\infty)}(t), $$ and the expectation of $\tau$ is $$ \int_0^\infty te^{-\lambda T}\mu e^{-e^{-\lambda T}\mu t}\ \mathsf dt = \frac{e^{\lambda T}}\mu.$$