= Solution
Let $K=\sum_iX_i$ and $\bar X=K/n$. The <likelihood> is $\theta^K(1-\theta)^{n-K}$, so for $0<K<n$ its unique maximizer is $\widehat\theta=K/n$. By invariance of <maximum likelihood estimation>,
$$
\boxed{\widehat\phi=\widehat\theta(1-\widehat\theta)=\frac{K(n-K)}{n^2}.}
$$
For $K=0$ or $n$, the strict parameter interval $(0,1)$ has no attained <likelihood> maximum; the usual extended estimate takes the limiting endpoint and gives $\widehat\phi=0$. Thus the displayed estimator is the conventional extended MLE on those samples, an important boundary qualification. This is the first estimator in <estimating Bernoulli variance>.
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