Example where $X$ and $Z$ are correlated, $Y$ and $Z$ are correlated, but $X$ and $Y$ are independent

$$X,Y,Z$$ are random variables. How to construct an example when $$X$$ and $$Z$$ are correlated, $$Y$$ and $$Z$$ are correlated, but $$X$$ and $$Y$$ are independent?

Intuitive example: $$Z = X + Y$$, where $$X$$ and $$Y$$ are any two independent random variables with finite nonzero variance.

• Or for that matter $Z=\max\{X,Y\}$. Commented Nov 30, 2020 at 7:49
• @GregMartin That is not guaranteed to work unless the supports of $X$ and $Y$ overlap Commented Nov 30, 2020 at 10:08
• @henry Nice point. But you don't even need them to overlap. Just that there exist values in $X$ which are smaller than some values in $Y$ and vice versa. E.g. discontinuous piece wise functions with no overlap can work. Commented Nov 30, 2020 at 11:32
• @DaleC - When I said "overlap" I meant that the minimum of each was less than the maximum of the other. Commented Nov 30, 2020 at 11:45
• A narrative example: two small kids from a soccer team trying to sneak into a movie by one sitting on the shoulders of the other and wearing a trench coat. Z is the height of the "man"; X and Y are the heights of the kids Commented Nov 30, 2020 at 23:40

Roll two dice.

X is the number on the first die, Z is the sum of the two dice, Y is the number on the second die

X and Z are correlated, Y and Z are correlated, but X and Y are completely independent.

(This is a concrete instance of the answer given by fblundun, but I came up with it before seeing their answer.)