proving that given independence of RV $X$ and $Y$,$\mathbb{E}[X|Y]=\mathbb{E}[X]$.

Question

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!!

Answer 1

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, $E(X\mid Y) =E(X)$.