 Solution

A simple birth process has rates $q_{n,n+1}=n\lambda$. Its pgf solves $G_t=\lambda z(z-1)G_z$, giving $G(z,t)=e^{-\lambda t}z/[1-(1-e^{-\lambda t})z]$. Thus from one ancestor $X_t$ is geometric with parameter $e^{-\lambda t}$ and mean $e^{\lambda t}$. From $i$ ancestors it is a sum of $i$ such variables, hence negative binomial with mean $i{e}^{\lambda t}$.