$G=X^Y$. Compute the expected value $\Bbb E(G\mid Y)$ and variance $V(G\mid Y)$.

Question

Let $X\sim \operatorname{U}(0,1)$ and $Y \sim \operatorname{Bi}(1,0.5)$ be independent random variables, $G=X^Y$. Compute the r.v. $\Bbb E(G\mid Y)$ and $V(G\mid Y)$.

I'm not sure how to compute the expected value of $\Bbb E(G(X,Y)\mid Y)$, could someone help me with this please? Thanks.

Answer 1

\begin{align} & \operatorname{E}(G\mid Y=0) = \operatorname{E}(X^Y\mid Y=0) = \operatorname{E}(1 \mid Y=0) = 1 \\ & \operatorname{E}(G\mid Y=1) = \operatorname{E}(X^Y\mid Y=1) = \operatorname{E}(X\mid Y=1) = \operatorname{E}(X) = \frac 1 2 \\ \text{Therefore } & \operatorname{E}(G\mid Y) = \begin{cases} 1 & \text{if } Y=0, \\ 1/2 & \text{if } Y=1, \end{cases} \\ \text{and so } & \operatorname{E}(G\mid Y) = \begin{cases} 1 & \text{with probability } 1/2, \\ 1/2 & \text{with probability } 1/2. \end{cases} \end{align}