When can E[Y(X)] equal E[Y(E(X))] for Gaussian variables?

[lode]

I have two random variables Y and X and Y is dependent of X, though X is not the only source of variability of Y. With fixed X=x, Y(x) follows gaussian law. X also follows gaussian law.

In what cases can I move from
E[ Y(X) ]

to
E[ Y(E(X))]

someone has any idea?
is there a text You recommend on the topic?
thanks..

 

[SW VandeCarr]

I have two random variables Y and X and Y is dependent of X, though X is not the only source of variability of Y. With fixed X=x, Y(x) follows gaussian law. X also follows gaussian law.

In what cases can I move from
E[ Y(X) ]

to
E[ Y(E(X))]

someone has any idea?
is there a text You recommend on the topic?
thanks..

I think you mean E(Y|X).

Can you combine X and R where R ( for 'residual') includes all other extraneous conditions on Y? If so, then write P(X)+P(R)-P(X)P(R)=P(X') assuming X and R are independent.

Then use Bayes' Theorem:

P(Y|X')=P(X'|Y)P(Y)/P(X')

The validity if this approach assumes that P(Y|X') includes all conditions on Y.

EDIT: You could of course simply find P(Y|X) the same way, but this does not give you as good a description of the behavior of Y when you know of other conditions on Y. E(Y|X') is the value of Y|X' when P(Y|X') is maximal.