I know if $g(X)=X^2$, the equality holds. Is there any other $g$? How can we generalize the case $g(X)=X^2$ (other than $g(X)=X^{2k}$?
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Every even function $g$, to begin with. Or $g=\mathrm 1_B$ where $B$ is such that $E(X;X\in B)=0$. And many more. – Did Feb 25 '14 at 17:27
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Got it. We just need to partition the domain of $g$ in to segment where $g$ is one to one. – bankrip Feb 25 '14 at 17:48
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Hmmm.... There might be cases when this is not even possible. – Did Feb 25 '14 at 17:50
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Can you give me a little bit more detail to think? How do you go about proving for the case $g$ is even? – bankrip Feb 25 '14 at 17:57
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Got something from the answer below? – Did Nov 19 '14 at 22:29
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Yes. Thank you! – bankrip Nov 19 '14 at 22:30
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Let $Y=g(X)$ and let $h$ denote some measurable function such that $E(X\mid Y)=h(Y)$. Assume that $g$ is even. Then $Y=g(-X)$, hence, the distribution of $X$ being symmetric, $(-X,Y)$ and $(X,Y)$ are identically distributed. Conditional expectations only depends on distributions hence $E(-X\mid Y)=h(Y)$, that is, $h(Y)=0$.
Did
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