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Problem

The covariance can be calculated this way

$ cov(X,Y) = E[XY] - E[X]E[Y] $ .

For a proof, check out this page.

Prove that

$ Var(X+Y) = Var(X) + Var(Y) + 2cov(X,Y) $ .

Variance-0

Solution

$ Var(X+Y) = E[{(X+Y)}^{2}] - {\left(E[X+Y]\right)}^{2} $

The expectation of sums is the sum of expectations.

$ Var(X+Y) = E[{(X+Y)}^{2}] - {\left(E[X] + E[Y]\right)}^{2} $
$ Var(X+Y) = E[X^2 + 2XY + Y^2] - {\left(E[X]\right)}^{2} - 2E[X]E[Y] - {\left(E[Y]\right)}^{2} $
$ Var(X+Y) = E[X^2] + 2E[XY] + E[Y^2] - {\left(E[X]\right)}^{2} - 2E[X]E[Y] - {\left(E[Y]\right)}^{2} $
$ Var(X+Y) = \left(E[X^2] - {\left(E[X]\right)}^{2} \right) + \left(E[Y^2] - {\left(E[Y]\right)}^{2} \right) + 2\left(E[XY] - E[X]E[Y] \right) $

Since $ Var(X) = E[X^2] - {\left(E[X]\right)}^{2} $,

$ Var(X+Y) = Var(X) + Var(Y) + 2cov(X,Y) $.
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