I saw the following question on another forum:
"Suppose that both height and weight of adult men can be described with normal models, and that the correlation between these variables is 0.65. If a man's height places him at the 60th percentile, at what percentile would you expect his weight to be?"
I see that someone at the forum in question has already pointed out that the question talks about the margins being normal (
height and weight ... can be described with normal models), not about bivariate normality and so the question doesn't have a single answer.
Clearly the answer would depend on the actual bivariate dependence relationship (the copula), which left me curious.
My question is:
Given normal margins and a specified population correlation ($\rho$, a Pearson correlation), is there a reasonably straightforward way to find bounds on $E(Y|X=x_q)$ given $X,Y$ both normal, with correlation $\rho$?
If there's an exact largest value and smallest value for the conditional expectation, that (and for preference, the circumstances in which each occurs*) would be good to know about.
* I have some strong suspicions about what those circumstances might be (i.e. the kind of dependence that might be involved; in particular, I expect a specific kind of degenerate distribution will give the bounds) but I haven't yet investigated that thought in any depth. (I figure someone is already likely to know it.)
Failing that, upper or lower bounds on both the largest and smallest values would be interesting.
I don't necessarily require an algebraic answer (some algorithm would do), though an algebraic answer would be nice.
Approximate or partial answers may be useful/helpful.
If nobody has good answers, I may have a go at it myself.