Additivity of the variance for independent random variables

Index:The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Additivity under independence

Theorem: The variance is additive for independent random variables:

p (X,Y) = p (X) p (Y) ⇒ V a r (X + Y) = V a r (X) + V a r (Y).(1)

$$(1)p(X,Y)=p(X)p(Y)\Rightarrow Var(X+Y)=Var(X)+Var(Y).$$

Proof: The variance of the sum of two random variables is given by

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

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

The covariance of independent random variables is zero:

p (X,Y) = p (X) p (Y) ⇒ C o v (X,Y) = 0.(3)

$$(3)p(X,Y)=p(X)p(Y)\Rightarrow Cov(X,Y)=0.$$

Combining (2) $(2)$ and (3) $(3)$, we have:

V a r (X + Y) = V a r (X) + V a r (Y).(4)

$$(4)Var(X+Y)=Var(X)+Var(Y).$$

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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: P130 | shortcut: var-add | author: JoramSoch | date: 2020-07-07, 06:52.