Solution
= Solution
The joint mass factorizes through $T=\sum_iX_i$, so $T$ is sufficient for $q$. If $\lambda=\sqrt q$, then
$$
E[X_1^2-X_1]=E[X_1(X_1-1)]=\lambda^2=q.
$$
Conditionally on $T$, $X_1\sim\operatorname{Bin}(T,1/n)$. Rao–Blackwell therefore gives the unbiased estimator
$$
E[X_1(X_1-1)\mid T]=\frac{T(T-1)}{n^2}.
$$
For $n\ge2$ the original estimator is not a <function> of $T$, so the variance reduction is strict.