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Let $V,W,Z$ independent Poisson($\lambda$)-distributed random variable. $\lambda>0$. $X:=V+W, Y:=V+Z$

how can i show, that X and Y are independent?

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  • $\begingroup$ I would be surprised if they were. If, say, $X=1000$, it makes it less likely that $V=0$, hence that $Y=0$. $\endgroup$ – Arnaud Mortier May 23 '18 at 22:22
  • $\begingroup$ Dumb question: $X$ and $Y$ both have $V$ in them so probably they aren't independent? Idk. Are there cases where you could have independence in a similar case? $\endgroup$ – BCLC May 26 '18 at 16:42
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$\newcommand{\c}{\operatorname{cov}}\newcommand{\v}{\operatorname{var}}$What would make you think they're independent? \begin{align} \c(X,Y) & = \c(V+W, V+Z) \\[10pt] & = \c(V,V) + \c(V,Z) + \c(W,V) + \c(W,Z) \\[10pt] & = \v(V)+0+0+0 \\[10pt] & = \lambda+0+0+0 \\[10pt] & = \lambda>0. \end{align}

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