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Addition Rule for Random Variables

  1. Oct 16, 2014 #1
    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,

    Best,

    Peter G.
     
  2. jcsd
  3. Oct 16, 2014 #2

    jfizzix

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    Let's say you have a joint probability distribution [itex]P(X,Y)[/itex].
    Then
    [itex]E(X)=\sum_{i}X_{i}P(X_{i})=\sum_{i,j}X_{i} P(X_{i},Y_{j})[/itex],
    and
    [itex]E(Y)=\sum_{j}Y_{j}P(Y_{j})=\sum_{i,j}Y_{j} P(X_{i},Y_{j})[/itex].
    From here, we can see that
    [itex]E(X)+E(Y)=\sum_{i,j}(X_{i}+Y_{j}) P(X_{i},Y_{j})= E(X+Y)[/itex].
    Hope this helps:)
     
  4. Oct 17, 2014 #3
    Thank you very much, jfizzix!
     
  5. Oct 17, 2014 #4

    jfizzix

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    No problem:)
     
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