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## Problem

The covariance can be calculated this way

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

Prove that

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

## 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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