Let X, Y and Z be random variables with finite means and finite variances. Suppose that the correlations ρ(X, Y ) and ρ(Y, Z) have unspecified positive values. Then without additional information on the distributions of X, Y and Z, the correlation ρ(X,Z) could possibly be positive, negative, or zero. In particular, we can find specific random variables X,Y,Z such that ρ(X, Z) is negative.
The question is to come up with a creative real-world example where two positively correlated pairs (X, Y ) and (Y, Z) of random variables could still give a negative correlation for (X,Z).
For example: In basketball, NBA basketball players could play as many as > 80 games per year, and their points scored over the year are recorded. For a randomly selected NBA basketball player, define the following random variables:
• Let X be the number of slam dunks made in a year.
• Let Y be the total number of points scored in a year.
• Let Z be the number of 3-pointers made in a year.
In this case, X,Y and Y,Z are both positively correlated, however X,Z is negatively correlated. I want to think of other creative examples like this (not related to other sports, please)