Assume that I have a random variable X~Exp(a)
And a random variable Y which is defined: Y = X + W, whereas
Y~Gamma(2,a), For Gamma distribution definition: Gamma distrubution
(r = 2, [Lambda]=a)
It is also known that if 2 random exponential variables with parameter a are independent, their sum is distributed as Gamma(2,a)
The question is: if I know X and Y, can I deduct that X and W are independent? Is there a sentence that works in this direction?
Thanks!
Take some $Z\sim Gamma\left(2,a\right)$ which is independent of $X$, and let $W = Z - X$. Then $Y = X + W = Z$, but
$$Cov\left(X,W\right) = Cov\left(X,Z\right) - Cov\left(X,X\right) = -Var\left(X\right).$$