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Variance of a sum: Var⁡(X  +  Y)\operatorname{Var}(X\; + \;Y) = ?

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Answer

  • Var⁡(X  +  Y)  =  Var⁡(X)  +  Var⁡(Y)  +  2  ⋅  Cov⁡(X,  Y)\operatorname{Var}(X\; + \;Y)\; = \;\operatorname{Var}(X)\; + \;\operatorname{Var}(Y)\; + \;2\; \cdot \;\operatorname{Cov}(X, \;Y).
  • Only when X and Y are uncorrelated (Cov = 0) does the covariance term vanish and Var⁡(X+Y)  =  Var⁡(X)  +  Var⁡(Y)\operatorname{Var}(X + Y)\; = \;\operatorname{Var}(X)\; + \;\operatorname{Var}(Y).
  • Crucial in portfolio theory (diversification reduces variance only if assets are not perfectly correlated), A/B testing (variance of the mean depends on within-subject correlation), and RL (variance-reduced estimators).
Check yourself — multiple choice
  • Var⁡(X+Y)  =  Var⁡(X)  +  Var⁡(Y)\operatorname{Var}(X + Y)\; = \;\operatorname{Var}(X)\; + \;\operatorname{Var}(Y) always
  • Var⁡(X+Y)  =  Var⁡(X)  +  Var⁡(Y)  +  2\operatorname{Var}(X + Y)\; = \;\operatorname{Var}(X)\; + \;\operatorname{Var}(Y)\; + \;2·Cov⁡(X,Y)\operatorname{Cov}(X, Y); only vanishes to sum when uncorrelated
  • Var⁡(X+Y)  =  Var⁡(X)\operatorname{Var}(X + Y)\; = \;\operatorname{Var}(X)·Var⁡(Y)\operatorname{Var}(Y)
  • Not defined

Variance of a sum: needs the covariance term unless uncorrelated.

#variance#probability

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