I am trying to compute the distribution of the following $$Z=\bigl(X+Y\bigl)^2$$ BUT I have that both $X$ and $Y$ are Nakagami with parameter $m$. (A Nakagami random variable is the square root of a Gamma random variable.) So the above is hard to derive in general ( I would have to take the convolution assuming the summands are independent..)
So my alternative is to solve $$W=X^2+Y^2$$ which is easier to compute as it will be the sum of two Gamma distributions with scale and shape parameter $m$ and is also Gamma distributed.
My question is, when can one argue that $W$ and $Z$ have the same distribution, is it when X and Y are uncorrelated then they are equivalent? or will they never be equivalent?