 Solution

The joint mass factorizes through $T={\sum }_i{X}_i$, so $T$ is sufficient for $q$. If $\lambda =\sqrt{q}$, thenConditionally on $T$, $X_1\sim \mathrm{Bin}(T,1/n)$. Rao–Blackwell therefore gives the unbiased estimatorFor $n\ge 2$ the original estimator is not a function of $T$, so the variance reduction is strict.