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# Why are expected value and variance additive? watch

1. 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?
2. (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
3. Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)

When X and Y are independent there is no covariance so simplifies down.
4. (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?
5. (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 ?

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Updated: February 16, 2016
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