TITLE: X,Y,Z are mutually independent random variables. Is X and Y+Z independent?
QUESTION [6 upvotes]: X,Y,Z are mutually independent random variables. Is X and Y+Z
  independent?

Please, give me a hint how to prove it?

REPLY [3 votes]: 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.