Law of Iterated Expectations:

E[y] = Ex [E [y | x]]

The notation Ex [.] indicates the expectation over the values of x. Note that E [y | x] is a function of x.

Covariance:

In any bivariate distribution,

Cov [x, y] = Covx [x, E[y | x]] = Integration (x - E[x]) E[y | x] fx (x)dx

(Note that this is the covariance of x and a function of x).

Can people explain this two theories in plain words?

What is the meaning of "the expectation over the value of x"?

What does the textbook mean by "this is the covariance of x and a function of (x)"? Why is the textbook saying the equation Cov [x, y] is the covriance of x and a function of x?