Addition Rule for Random Variables

[Peter G.]

Hi,

I am having a hard time understanding why the Addition Rule for two Random Variables holds even when the random variables are dependent.

Essentially: why is E(X+Y) = E(X) + E(Y) when X and Y are dependent random variable?

Given the two variables are dependent, if X happens to take on a value x, for example, doesn't that change the probability distribution of Y and, thus, affect its expected value?

I hope I made my doubt clear,
Peter G.

 

[jfizzix]

Let's say you have a joint probability distribution $P(X,Y)$.
Then
$E(X)=\sum_{i}X_{i}P(X_{i})=\sum_{i,j}X_{i} P(X_{i},Y_{j})$,
and
$E(Y)=\sum_{j}Y_{j}P(Y_{j})=\sum_{i,j}Y_{j} P(X_{i},Y_{j})$.
From here, we can see that
$E(X)+E(Y)=\sum_{i,j}(X_{i}+Y_{j}) P(X_{i},Y_{j})= E(X+Y)$.
Hope this helps:)

 

[Peter G.]

Thank you very much, jfizzix!

 

[jfizzix]

No problem:)