random variables independence

Question

I need to check if Z and W are dependent or not.

X,Y ~ Exp(2)

Then I define: Z=X-Y , W=X+Y.

Now How I can check that Z and W are dependent or not ?

I know from the teory that I should show that f(z,w) = f(z)f(w) , but I dont see how I do it here.

Thanks for help.

The observation that $$P(Z\lt W)=1$$ should be nearly enough to conclude. That, or computing the density of $(Z,W)$ by the standard method. — Did, Mar 13 '15 at 15:07

score 3 · Accepted Answer · answered Mar 13 '15 at 15:47

3

If $Z$ and $W$ are indeed independent then e.g. $P(Z>1,W<1)=P(Z>1)P(W<1)>0$.

This contradicts that $P(Z<W)=1$ as mentioned in the comment of @Did.

answered Mar 13 '15 at 15:47

drhab

1 Answers1