Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 1 / 18C / ii / Solution

Independence gives $\mathbb{E}[X_1(1-X_2)]=\theta (1-\theta )=\varphi$, proving unbiasedness. Conditional on $K=k$, all binary sample strings with $k$ ones are equally likely. ThereforeThe Rao-Blackwell theorem, or the tower property and total variance decomposition directly, yields the improved estimatorIt is unbiased by conditional expectation and is a function of the requested MLE. Its variance is strictly smaller, sinceFor $0<\theta <1$, the event $K=1$ has positive probability. On it, $X_1(1-X_2)$ has probability $1/n$ of being one, strictly between zero and one for $n\ge 2$. Thus the final expectation is strictly positive. This proves strict variance reduction, not merely a weak inequality.