Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 8 / b / iii / Solution

For $n\ge 2$, $X_2+\cdots +X_n$ is independent of $X_1$ and has a Poisson distribution with parameter $(n-1)\lambda$. Part (ii) therefore givesIts probability at zero isFor $n=1$ the event being conditioned on is $X_1=1$, so the answer is $0$, also matching the displayed expression. The cancellation of $\lambda$ reflects the equal rates of all $n$ counts, not an approximation or large-$n$ limit.