# Math Help - I have problems of understanding two statistical concepts

1. ## I have problems of understanding two statistical concepts

The followings are the definitions of Law of Iterated Expectations and Covariance from textbook.

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?

2. Originally Posted by Real9999
The followings are the definitions of Law of Iterated Expectations and Covariance from textbook.

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.
First we need Bayes' theorem:

$p(x,y)=p(x|y)p(y)=p(y|x)p(x)$

Then

$\displaystyle E(y)= \int_x \int_y y p(x,y)\;dydx$

and:

$\displaystyle E_x(E(y|x))=\int_x \left(\int_y y p(y|x)\;dy \right) p(x)\;dx$

.............. $\displaystyle =\int_x \int_y \left( y \frac{p(x|y)p(y)}{p(x)} \right) p(x)\;dy dx= \int_x \int_y y p(x,y)\;dydx$

etc.

CB

3. Originally Posted by Real9999

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?
Like the other just write out the definitions and apply Bayes' theorem and some algebra.

CB

4. Thank you so so so much XD