Sample Question #278 (statistics/probability theory)

1) If random variables X and Y are negatively correlated, and Y and Z are negatively correlated, are X and Z positively or negatively correlated?

2) Give me an example of three random variables that display the behavior described in part 1 above.

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ANSWER

1) We cannot say anything about the correlation between X and Z.

2) Actually I screwed up on the wording of this part [aren’t you glad you’re reading this comment?]. In the actual interview I got part 1 right, so the interviewer asked me to give him an example of X, Y, Z that were mutually negatively correlated. One such example is the traffic light; do you know why?

Ex A: X=-5,2,3 Y=1,0,-1 Z=2,-5,3.Each r.v. assumes respective value with probability 1/3.Cov(X,Y)=-8/3<0, Cov(Y,Z)=-1/3<0, Cov(X,Z)=-11/3<0.Ex B: Now set Z=X. Then Cov(X,Y)<0, Cov(Y,Z)<0, Cov(X,Z)>0.

How can we prove that X, Y, Z can not have relative correlations 0.9, 0.2, 0.7?

Noname says:

"How can we prove that X, Y, Z can not have relative correlations 0.9, 0.2, 0.7?"

Now that’s a really good question. I saw it somewhere before. I just hope I’m not asked that at an interview!

Hey, quantyst, how did you come up with that one? 😉

Thanks for all your helpful comments to this blog.