X,Y,Z are mutually independent random variables. Is X and Y+Z independent?

Please, give me a hint how to prove it?

  • $\begingroup$ 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. $\endgroup$ – mirastok Sep 6 '14 at 16:38
  • $\begingroup$ Look into the mathematical definition of independent variables $\endgroup$ – zed111 Sep 6 '14 at 16:59

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.


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