independent Poisson-distribution random variable

Question

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?

Answer 1

$\newcommand{\c}{\operatorname{cov}}\newcommand{\v}{\operatorname{var}}$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}