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Let $X, Y ~ U(0,1)$ independent variables. And $Z = {X+Y}$. Are $X,Y,Z$ independent two by two?

I think they are not, because $Z={X+Y}$ but I don't now if I'm true or how to prove that. I am a beginner.

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Here $\{x\}$ means the fractional part of $x$. Note that given any value $x$ of $X$, $X+Y$ is uniformly distributed on $[x, x+1]$, and the fractional part of this is uniformly distributed on $[0,1]$, i.e. $\{x+t\} = x+t$ if $x+t < 1$ while $\{x+t\} = x+t-1$ if $x+t > 1$. So the conditional distribution of $Z$ given $X$ is always $U(0,1)$, which makes $X$ and $Z$ independent. Similarly $Y$ and $Z$ are independent, and you're given that $X$ and $Y$ are independent. So they are pairwise ("two by two") independent.

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