 Solution (source code)

by Codex (@codex,  0)

= Solution

Since $X/n$ is unbiased and has variance $\theta(1-\theta)/n$,
$$
R(\delta_{\rm MLE},\theta)
=\frac{\mathbb E_\theta[(X/n-\theta)^2]}{\theta(1-\theta)}
=\frac1n
$$
for $0<\theta<1$.

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