Okey imagine that you have two random variables $ X,Y \sim Bin(1,\frac{1}{2})$, which are independent. Define $Z_1, Z_2$ with $ Z_1 := | X - Y | $ and $Z_2 = X + Y$. Show that $Z_1$ and $Z_2$ are not independent.
I already solved this exercise. So why I ask this question? A friend of mine asked me today if his solution is correct. To be honest I didn't know, so I've deciced to ask you. He said that:
it holds that $ Z_1 + Z_2 = 2X $ ,$\mbox{ }$ $ Z_1 - Z_2 = - 2Y $ if $ X \geq Y $ . and $Z_1+Z_2 =2Y$ , $Z_1 - Z_2 = -2X $ if $ X < Y $. Then he said that we get $Z_1 = |Z_2 - 2X | = | Z_2 - 2Y |$ so $Z_1 = \pm(Z_2 - 2X) = \mp (Z_2 - 2Y)$, which show that $Z_1$ and $Z_2$ are not independent? So is this really correct?
His argument is not correct. I don't see how he got $Z_1=|Z_2-2X|=|Z_2-2Y|$. Even if this is true (in the last line) how did he jump to the conclusion that $Z_1$ and $Z_2$ are not independent?