Let $X$ and $Y$ be two i.i.d. RVs with values in $\mathbb{R}^n$ and zero expectation. Suppose $Z=\alpha X+ \beta Y$ for non-zero constants $\alpha$ and $\beta$. I was wondering if the following formula is correct, and if someone can provide hints how to prove (or reject) them:
\begin{eqnarray} \mathbb{E}[X|Z]&=&\frac{\alpha}{\alpha^2+\beta^2} Z \ \ \ \textrm{and} \ \ \ \mathbb{E}[Y|Z]&=&\frac{\beta}{\alpha^2+\beta^2} Z \end{eqnarray}
I tried to invoke symmetry arguments to justify these expressions but couldnt succeed. The motivation is that they hold true at least for some distributions (e.g. Normal) and one might think to generalize them.
Hint: $E(E(X|Y)) = E(X) = \mu$. Consider your claim $$ E(E(X|Z)) = \frac{\alpha}{\alpha^2 + \beta^2} E(Z) = \frac{\alpha (\alpha + \beta) \mu}{\alpha^2 + \beta^2} \stackrel{?}{=} \mu $$