Variance of Sums of Random Variables

Compute the variance of sums, differences, and linear combinations of random variables that may be correlated, using covariance terms.

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The Variance of a Sum

For random variables XX and YY that may not be independent, the variance of their sum depends on how the two variables vary together:

Var⁡(X+Y)=Var⁡(X)+Var⁡(Y)+2Cov⁡(X,Y).\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X,Y).

This generalizes the independent case. When Cov⁡(X,Y)=0\operatorname{Cov}(X,Y) = 0, the formula collapses to Var⁡(X+Y)=Var⁡(X)+Var⁡(Y).\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y).

For example, if Var⁡(X)=4,\operatorname{Var}(X) = 4, Var⁡(Y)=9,\operatorname{Var}(Y) = 9, and Cov⁡(X,Y)=3,\operatorname{Cov}(X,Y) = 3, then

Var⁡(X+Y)=4+9+2(3)=19.\operatorname{Var}(X+Y) = 4 + 9 + 2(3) = 19.
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