= Solution
For $n\geq2$, $X_2+\cdots+X_n$ is independent of $X_1$ and has a <Poisson distribution> with parameter $(n-1)\lambda$. Part (ii) therefore gives
$$
X_1\mid(X_1+\cdots+X_n=n)\sim\operatorname{Bin}(n,1/n).
$$
Its probability at zero is
$$
\boxed{\mathbb P(X_1=0\mid X_1+\cdots+X_n=n)=\left(1-\frac1n\right)^n\quad(n\geq1)}.
$$
For $n=1$ 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.
Back to article page
