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Why are expected value and variance additive? watch

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    So I've just been learning about expected value and variance for discrete random variables and these 2 equations:

    1- E(X+Y) = E(X) + E(Y) for any random variables X,Y
    2- Var(X+Y) = Var(X) + Var(Y) for independent X,Y

    (1) makes sense for independent X,Y; but how do we know for sure that it's true for all random variables X,Y?

    And why is (2) true?

    Is there a simple algebraic (or other) proof of these?
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    (Original post by Pronged Lily)
    So I've just been learning about expected value and variance for discrete random variables and these 2 equations:

    1- E(X+Y) = E(X) + E(Y) for any random variables X,Y
    2- Var(X+Y) = Var(X) + Var(Y) for independent X,Y

    (1) makes sense for independent X,Y; but how do we know for sure that it's true for all random variables X,Y?

    And why is (2) true?

    Is there a simple algebraic (or other) proof of these?
    Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y )....

    see pages 2,3 here:

    http://www.cs.kent.edu/~jin/GM09/GM_Probability.pdf
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    Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)

    When X and Y are independent there is no covariance so simplifies down.
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    (Original post by the bear)
    Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y )....

    see pages 2,3 here:

    http://www.cs.kent.edu/~jin/GM09/GM_Probability.pdf
    Sorry if this is a stupid question but what does P(X ∧ Y) mean?
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    (Original post by Pronged Lily)
    Sorry if this is a stupid question but what does P(X ∧ Y) mean?
    not sure... google did not help much either... TeeEm might know ?

    :holmes:
 
 
 
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