hi i really need some help with this question. I need to prove given that two random variables $X$ and $Y$ are independent,then $X$ and $Y$ are mean-independent.(i.e $\mathbb{E}[X|Y]=\mathbb{E}[X]$. How do i go about this? THANKS!!
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Do you mean $\mathbb{E}[X|Y]=\mathbb{E}[X]$? – msm Sep 13 '16 at 09:59
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Use the fact that when $X$ and $Y$ are independent, the conditional distribution $f_{X/Y}=f_{X}$ to get $E[X/Y]=E[X]$ – nemo Sep 13 '16 at 10:10
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Yes thats what i meant. sorry about that. Can you help me out? – paapa essel Sep 13 '16 at 10:10
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o okay, i ll try that out and see what comes up.thanks!!!! – paapa essel Sep 13 '16 at 10:11
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If $X$ and $Y$ are independent, then, for any Borel measurable set $A$, $X$ and $(Y\in A)$ are also independent. Therefore, \begin{align*} \int_{Y\in A}E(X\mid Y) dP &= \int_{Y\in A} X dP\\ &=\int_{\Omega} X \,\mathbb{I}_{Y\in A}dP\\ &=E(X \,\mathbb{I}_{Y\in A})\\ &=E(X) E(\mathbb{I}_{Y\in A})\\ &=E(E(X)\,\mathbb{I}_{Y\in A})\\ &=\int_{\Omega}E(X)\,\mathbb{I}_{Y\in A} dP\\ &=\int_{Y\in A} E(X) dP. \end{align*} That is, $E(X\mid Y) =E(X)$.
Gordon
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