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:)

 

[blue_raver22]

Hello Peter,

The Addition Rule for Random Variables holds even when the variables are dependent because of the linearity of expectation. This means that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of their dependency.

To understand this concept, let's consider a simple example. Let X and Y be two dependent random variables, where X represents the number of red balls in a bag and Y represents the number of blue balls in the same bag. The total number of balls in the bag is given by X+Y. Now, if we want to find the expected value of the total number of balls, E(X+Y), we can break it down into two parts: E(X) and E(Y).

Since X and Y are dependent, the probability distribution of Y will change depending on the value of X. However, when we calculate E(Y), we are considering all possible values of Y, not just one specific value. This means that the change in probability distribution due to the value of X is already accounted for in E(Y). Therefore, when we add E(X) and E(Y) together, we are essentially taking into account the change in probability distribution of Y caused by X.

In other words, the expected value of a sum of random variables takes into account the dependencies between the variables. This is why the Addition Rule holds even when the variables are dependent.

I hope this explanation helps clarify your doubt. Let me know if you have any further questions.

Best,