Suppose that $X$ has a binomial distribution with parameter $N=1$ and $p=1/2$. Y, which is independent of $X$, has a normal distribution with mean $\mu$ and variance 1. Consider the estimator $\mu$ of the form $W_1 = Y + 2X -1$. (Please see my work after parenthesis.)
(a.) Is $W_1$ unbiased? (Yes, because $E(W_1) = E(Y) + 2E(X) - 1 = \mu$)
(b.) What is the variance of $W_1$? ($Var(W_1)=Var(Y)+4Var(X)=1 + 4p(1-p)$)
(c.) Consider the estimator $W_2 = E[W_1|Y]$. Is $W_2$ unbiased? How does its variance compare to that of $W_1$? (I am not sure how to deal with conditional expectation here.)
Thank you very much.
$W_2 = E(Y\lvert Y) + 2E(X\lvert Y) - E(1\lvert Y) = Y + 2E[X] -1 = Y$ so $W_2$ is unbiased and its variance is equal to 1.