Variance of the sum of two random variables

Index:The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Variance of a sum

Theorem: The variance of the sum of two random variables equals the sum of the variances of those random variables, plus two times their covariance:

V a r (X + Y) = V a r (X) + V a r (Y) + 2 C o v (X,Y).(1)

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

Proof: The variance is defined in terms of the expected value as

V a r (X) = E [(X − E (X)) 2].(2)

$$(2)Var(X)=E[(X-E(X))2].$$

Using the linearity of the expected value and the definition of the covariance, we can derive (1) $(1)$ as follows:

V a r (X + Y) (2) = E [((X + Y) − E (X + Y)) 2] = E [([X − E (X)] + [Y − E (Y)]) 2] = E [(X − E (X)) 2 + (Y − E (Y)) 2 + 2 (X − E (X)) (Y − E (Y))] = E [(X − E (X)) 2] + E [(Y − E (Y)) 2] + E [2 (X − E (X)) (Y − E (Y))] (2) = V a r (X) + V a r (Y) + 2 C o v (X,Y).(3)

$$(3)Var(X+Y)=(2)E[((X+Y)-E(X+Y))2]=E[([X-E(X)]+[Y-E(Y)])2]=E[(X-E(X))2+(Y-E(Y))2+2(X-E(X))(Y-E(Y))]=E[(X-E(X))2]+E[(Y-E(Y))2]+E[2(X-E(X))(Y-E(Y))]=(2)Var(X)+Var(Y)+2Cov(X,Y).$$

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Sources:

• Wikipedia (2020): “Variance”; in: Wikipedia, the free encyclopedia, retrieved on 2020-07-07; URL: https://en.wikipedia.org/wiki/Variance#Basic_properties.

Metadata: ID: P128 | shortcut: var-sum | author: JoramSoch | date: 2020-07-07, 06:10.