Prove $\mathbb E(X∣Y)=0$ for $2$ variables

Question

I was looking for an example of two dependent random variables in which
$\mathbb E(X|Y)=\mathbb E(X)$

I found this example:
$X∼U[−1,1]$ and $Y=X^2$

How can I prove that $\mathbb E(X∣Y)=0$?

thanks!

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Jan 10 '20 at 08:27

score 1 · Answer 1 · answered Jan 10 '20 at 08:24

1

$E(XI_{{X^{2}} \leq x})=0$ for each $x$ because $X$ has same distribution as $-X$. This implies that $E(XI_A)=0$ for any $A \in \sigma (Y)$ and hence $E(X|Y)=0$.

answered Jan 10 '20 at 08:24

Kavi Rama Murthy

311,013

Thanks for your quick answer!! I can understand why in each symmetrical section the expectation of X is zero, but why E(X|Y)=0 also? Maybe I missed something.. – Nofar Piri Jan 10 '20 at 08:38

score 0 · Answer 2 · answered Jan 10 '20 at 11:07

0

Hint: $E(X|X^2) = E(-X|X^2) = -E(X|X^2)$

answered Jan 10 '20 at 11:07

Eugene Sirkiza

710

Thank you! It was very helpful – Nofar Piri Jan 10 '20 at 15:17

Prove $\mathbb E(X∣Y)=0$ for $2$ variables

2 Answers2