Question: Let X₁, X₂, X₃ and X₄ be independent binary random variables taking the value 1 with probability p and 0 with probability 1-p. We define two estimators:

\[{{\hat{p}}_{1}}=\bar{X}=({{X}_{1}}+{{X}_{2}}+{{X}_{3}}+{{X}_{4}})/4\] and
\[{{\hat{p}}_{2}}=\bar{X}/2+1/4\]

For what values of p is the mean squared error (MSE) of \[{{\hat{p}}_{2}}\] smaller than that of \[{{\hat{p}}_{1}}\] ?