Expectation value of the sum of two random variables

[jg370]

Homework Statement

The expectation value of the sum of two random variables is given as:

\langle x + y \rangle = \langle x \rangle + \langel y \rangle

My textbook provides the following derivation of this relationship.

Suppose that we have two random variables, x and y. Let p_{ij} be the probability that our measurement returns x_{i} for the value of x and y_{j} for the value of y. Then the expectation value of the sum of x+y is:

\langle x + y \rangle = \sum\limits_{ij} p_{ij} (x_i + y_j) =\sum\limits_{ij} p_{ij} x_i + \sum\limits_{ij} p_{ij} x_j

Then I am given the following statement:

But \sum\limits_j p_{ij} = p_i is the probability that we measure x_i regardless of what we measure for y, so it must be equal to p_i. Similarly, \sum\limits_i p_{ij} = p_j, is the probability of measuing y_i irrespective of what we get for x_i.

Homework Equations

The difficulty I have with this statement is that I do no see how \sum\limits_j p_{ij} can be equal to p_i.

The Attempt at a Solution

Summing over j, we should have (p_{i1} + p_{i2},+ ... p_{in}). Now, is this equal to p_i.

And similarly how can \sum\limits_i p_{ij} can be equal to p_j

I am hopefull that someone can clear this up for me.

Thank you for your kind assitance.

jg370[/quote]

 

[lanedance]

so you have 2 discrete random variables X & Y, with a joint distribution, pij
p_{ij} = P(X=x_i, Y = y_j)

the expectation is given by:
<X+Y> = \sum_{ij} p_{ij} x_i y_j =

By definition, the marginal probabilities are
P(X=x_i) = \sum_{j} p_{ij} = p_i
P(Y=y_j) = \sum_{i} p_{ij} = p_j

If the variables are independent then you have the further conidtion that
p_{ij} = P(X=x_i, Y = y_j) = P(X=x_i)P(Y = y_j) = p_i p_j