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Suppose that $X$ and $Y$ are independent random variables, $X$ is uniformly distributed on $[0,1]$, and $Y$ is uniformly distributed on $\{ 1,2,3\}$. How to get $E((X+Y)^2 | Y)$?

It seems to me that the conditional density $p_{X|Y}(x|y)$ is equal to $p_X(x)$, so what is the role of $Y$ here?

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    Since $Y$ can have only three possible values, you can just treat each of the three cases separately, i.e. $$E[(X+Y)^2\,|\,Y] = \begin{cases} E[(X+1)^2] & \text{if }Y=1 \\ E[(X+2)^2] & \text{if }Y=2 \\ E[(X+3)^2] & \text{if }Y=3 \end{cases}$$ Hopefully this may help you come up with a general rule, but if not, a case-by-case answer is still a valid answer.2011-08-28

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