Conditional Mean: Does the Equation Hold in General?

[EconMax]

In a paper, I have found this relationship:

E(X|Y<y)=Cov(X,Y)*E(Y|Y<y)/Var(Y)

It seems to me that the previous equation holds if, for instance, X=aY+Z with Z and Y independent and a non zero.

It also holds if (X,Y) is a bivariate normal (with non zero correlation).

But does it hold in general?

I think the answer is no. Because, trivially, if X and Y are independent, the equation is wrong.

Am I correct?

 

[statdad]

It doesn't have to hold if the variables are independent: take X, Y as independent, so that $$cov(X,Y) = 0$$, with $$E(X) = a \ne 0$$. Then the left side is non-zero, the right side is zero.

 

[EconMax]

True. In the case that X=aY+Z, it does not hold also if Corr(Y,Z) is different from zero and/or E(Z) is different from zero (I forgot to mention this condition before).