The question is about a particular exercise.
Let X,Y be independent random variables with exponential distribution where both have the same parameter $\lambda$. Define $Z_1:=X_1+X_2$. Calculate the density of $Z_1$. Also define $Z_2:=\frac{X_1 X_2}{X_1+X_2}$.
$Z_3:=\frac{X_1^2}{X_1+X_2}$
$Z_4:=\frac{X_2}{X_1+X_2}$
Are $Z_1,Z_2$ independent?
Are $Z_1,Z_3 $independent?
Are $Z_1,Z_4$ independent?
My Approach: I calculated the densitiy of Z$_1$by using the convolution formula, but now I'm struggling to show that $Z_1$ and $Z_4$ are independent. (I got a hint that these two are independent.)