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I will have a Statistics course soon, so I decided to start a Probability Theory class myself and learn the "basics" first before I start the statistics. I reached the chapter Conditioning on Random Variable's and there was this question I find very difficult to solve. The question is;

Let X and Y be independent, continuous random variables with expected value and variance µX,σ2 X and respectively µY ,σ2 Y . Determine/ Calculate; E[(X + Y )X |Y ].

I know how to integrate over already probability density functions and then calculate the value's, but with the conditioning on a stochastic it becomes increasingly more difficult! Could someone help me out or give me any hints?

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    $\begingroup$ Two useful facts: (a) $\sigma_X^2=E[X^2]-\mu_X^2$ and (b) if $X$ and $Y$ are independent then $E[XY]=\mu_X \mu_Y$ $\endgroup$
    – Henry
    Dec 5 '16 at 19:37
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$$\begin{align}\mathsf{E}[(X+Y)X|Y]&=\mathsf{E}[(X^2+XY)|Y]\\ &\stackrel{(1)}{=}\mathsf{E}[X^2|Y]+\mathsf{E}[XY|Y]\\ &\stackrel{(2)}{=}\mathsf{E}[X^2]+\mathsf{E}[X]\mathsf{E}[Y]\\ &\stackrel{(3)}{=}\sigma^2_X+\mu^2_X+\mu_X\mu_Y \end{align}$$

where $(1)$ is due to the linearity of expectation, $(2)$ is because of independence of $X$ and $Y$, and $(3)$ according to definition $\sigma^2_X=\mathsf{E}[X^2]-\mathsf{E}[X]^2$.

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