I'm stuck on the following problem:
$X_i \sim N(\theta ,1) , \theta \in [0,1] , i \in \{0,1\}$
Prove $E[(0.5(X_1 + X_2)-\theta)^2]=0.5 $
Note: $X_1$ and $X_2$ are independent
My work so far:
\begin{align} &= E[(0.5(X_1 + X_2)-\theta)^2] \\& = E[0.25(X_1 + X_2)^2-(X_1 + X_2)\theta + \theta^2] \\ & = E[0.25(X_1^2+2X_1X_2+X_2^2)]-\theta^2 \\ & = \frac14(2+2\theta^2+2E[X_1X_2])-\theta^2\end{align}
To evaluate $E[X_1^2]$ I used the fact that: $Var[X]=E[X^2]-E[X]^2$
Now my questions is how to evaluate what $E[X_1X_2]$ is? My intuition is that it should be $1+\theta^2$ as $X_1$ and $X_2$ are identical but that appears to be wrong.