Examine whether $U(X_1,X_2)$ is an unbiased estimator.

Question

I am confused right now in (ii) part. I am thinking it as joint pdf and i am calculating expectation using double summation. Can someone tell me how to take expectation of this statistic.

Answer 1

\begin{align} \operatorname E(U(X_1,X_2) = {} & \frac 1 {2n}\Pr(X_1=1\ \&\ X_2=0) + \frac 1 {2n}\Pr(X_1=0\ \&\ X_2=1) \\[10pt] & {} + 0\Pr(X_1=0=X_2) + 0\Pr(X_1=1=X_2). \end{align}