# correlation of r1 and r2 is x. Probability of r1 > r2?

Just a quick probability interview question.

If the correlation of two variables r1, r2 is x. What's the probability that a sample of r1 is greater than a sample of r2?

let me update the question. This is what the interviewer originally asked: the stock moves every day (r1 in my model) and you have a predictor (r2 in my model) that predicts the stock return with correlation of x. You make trades according to this predictor. What's the portion of your winning trades?

let's add a stronger assumption that they have the same distribution

• You're not giving us a lot to go on. – rolando2 Feb 11 '15 at 3:11
• @rolando2 I updated the original question, in case I mismodel the original question. – Vendetta Feb 11 '15 at 3:17
• More than a smidgeon. – wolfies Feb 11 '15 at 4:41
• Are we assuming equal mean and variance, and the same distribution, for the two variables? – Jeremy Miles Feb 11 '15 at 4:43
• @JeremyMiles yes, let's assume that. – Vendetta Feb 11 '15 at 16:26

The two quantities in the question have little relationship to one another, because correlation is invariant under changes of scale and location whereas stochastic dominance is not.

I will give a set of examples in which all possible combinations $(\rho, p)$ of correlation coefficient $\rho$ and chance $p=\Pr(r_1\gt r_2)$ are realized. All correlations are achievable by certain families of bivariate random variables $(X,Y)$, supported on a subset $[0,1]\times[0,1]$, with continuous margins, each of which is supported on $(0,1)$. (For instance, let them be uniform in an ellipse $x^2+y^2-2\rho x y \le 1-\rho^2$ inscribed within that square. These distributions have correlation coefficient $\rho$.)

By setting $r_1 = X+\mu$ and $r_2=Y$, $\Pr(r_1\gt r_2)$ ranges continuously from $0$ to $1$ as $\mu$ ranges from $-1$ to $1$ without changing the correlation. Therefore this chance can be any value between $0$ and $1$ inclusive.

Although the marginal distributions in the specific example I gave are identical in each case, they do not have to be identical. When they are not, there will be nontrivial lower and upper bounds on the correlations which can actually be attained, but the same (constructive) argument shows that knowing the correlation tells us (to within an arbitrary accuracy) nothing about $\Pr(r_1 \gt r_2)$.

Formulating the question in terms of "samples" does not change the answer, because a sufficiently large sample will have the characteristics of the parent distribution.

The updated question is different: the "portion of winning trades" appears to be the chance that $r_1 \gt 0$. This is easily addressed using the same methods; I leave that answer to the reader.

• Much better example than in my comment... although I was about to jot down something similar, honestly :) – P.Windridge Feb 11 '15 at 19:21