 Solution

Let $K={\sum }_i{X}_i$ and $\bar{X}=K/n$. The likelihood is ${\theta }^K(1-\theta {)}^{n-K}$, so for $0<K<n$ its unique maximizer is $\hat{\theta }=K/n$. By invariance of maximum likelihood estimation,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 $\hat{\varphi }=0$. 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.