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[See Solution] (1.5 marks) Let X 1 , X 2 , X 3 and X 4 be independent binary random variables taking the value 1 with probability p and 0 with probability 1-

Question: (1.5 marks)

Let X 1 , X 2 , X 3 and X 4 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 \[{{\hat{p}}_{1}}\] ?

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