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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=

*a*Y+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?