Let $X_1$ and $X_2$ and $Z$ denote independent, real valued random variables and $$Pr(Z=0) = 1-Pr(Z=1) = \alpha$$ for some $0<\alpha<1$ Define
$Y$ = $X_1 $ if $Z=0$
= $X_2$ if $Z=1$
(a) Suppose that $E(X_1)$ and $E(X_2)$ exist, does it follow that E(Y) exists? (b) Assume that $E(|X_1|) < \infty $ and $E(|X_2|) < \infty $. Find $E(Y|X_1)$
This is NOT homework. This is from selfstudy. I would appreciate any kind of help. Intuitively, I feel that the answer for (a) is $E(X_1).\alpha+(1-\alpha).E(X_2)$ but I don't have any rigorous solution. For (b) I am totally confused not knowing even how to start. I don't even understand why the "$ < \infty $" is necessary.
I appreciate it.