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$X,Y,Z$ 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?

develarist
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Charlotte
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    A good example is given here: https://stats.stackexchange.com/questions/498040/is-this-possible-that-corx-y-0-99-cory-z-0-99-but-corx-z-0 – markowitz Nov 29 '20 at 13:12

2 Answers2

49

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

fblundun
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    Or for that matter $Z=\max\{X,Y\}$. – Greg Martin Nov 30 '20 at 07:49
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    @GregMartin That is not guaranteed to work unless the supports of $X$ and $Y$ overlap – Henry Nov 30 '20 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. – Dale C Nov 30 '20 at 11:32
  • @DaleC - When I said "overlap" I meant that the minimum of each was less than the maximum of the other. – Henry Nov 30 '20 at 11:45
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    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 – Dancrumb Nov 30 '20 at 23:40
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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.)

Alsee
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