E[E[Y|X]]

Why E[E[Y|X]]=sum E[Y|X]*p(x)?

When we calculate E[Y|X], we calculate sum y*p(x,y)/p(x).
It means y multiply conditional probability p(x|y).

Why E[E[Y|X]] not E[Y|X]*p(y)?
Why the probability of E[Y|X] is p(x)?
I think it should be p(y), because the object of expected value should be Y like E[Y|X],shouldn't it?
How can it become X?

 

Perhaps notation is getting you bogged down a bit here.

When you write E[E[Y|X]], the two expectations are being taken over different variables. The inner expected value E[Y|X] is being taken over Y, and the outer expected value is being taken over X.

This is because E[Y|X] is a random variable in terms of X. i.e. we can write the expected value of Y for a particular value of X as E[Y|X=x], which is clearly a function of x. Indeed we can define f(x)=E[Y|X=x]. Then the random variable E[Y|X] is f(X). So when we write E[E[Y|X]] this is E[f(X)], which is clearly an expectation with respect to different values X can take.

Hope that clears things up.

 

Thank you!It is super clear!