X,Y,Z are mutually independent random variables. Is X and Y+Z independent?
Please, give me a hint how to prove it?
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mirastok
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I took undegraduate course in probability, but it looks like I forgot a lot. I know basic distributions, understand linearity of expectation, and so on. – mirastok Sep 06 '14 at 16:38
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Look into the mathematical definition of independent variables – zed111 Sep 06 '14 at 16:59
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Based on this answer,
\begin{gather*} \mathbb{E}[e^{i(X+Y,Z)\cdot(s,t)}] = \mathbb{E}[e^{is(X+Y) + tZ}] \stackrel{*}= \mathbb{E}[e^{isX}]\mathbb{E}[e^{isY}]\mathbb{E}[e^{itZ}] = \mathbb{E}[e^{is(X+Y)}]\mathbb{E}[e^{itZ}], \end{gather*} where $*$ follows by assumption. I have assumed the expectations are well-defined.
Edit: note that this is different from $\mathbb{E}[e^{is(X+Y+Z)}] = \mathbb{E}[e^{is(X+Y)}]\mathbb{E}[e^{isZ}]$, which does not imply independence.
