$X,Y,Z$ 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?


2 Answers 2


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

  • 4
    $\begingroup$ Or for that matter $Z=\max\{X,Y\}$. $\endgroup$ Nov 30, 2020 at 7:49
  • 3
    $\begingroup$ @GregMartin That is not guaranteed to work unless the supports of $X$ and $Y$ overlap $\endgroup$
    – Henry
    Nov 30, 2020 at 10:08
  • $\begingroup$ @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. $\endgroup$
    – Dale C
    Nov 30, 2020 at 11:32
  • $\begingroup$ @DaleC - When I said "overlap" I meant that the minimum of each was less than the maximum of the other. $\endgroup$
    – Henry
    Nov 30, 2020 at 11:45
  • 3
    $\begingroup$ 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 $\endgroup$
    – Dancrumb
    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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.