Law of total covariance

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In probability theory, the law of total covariance,[1] covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then

[math]\displaystyle{ \operatorname{cov}(X,Y)=\operatorname{E}(\operatorname{cov}(X,Y \mid Z))+\operatorname{cov}(\operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z)). }[/math]

The nomenclature in this article's title parallels the phrase law of total variance. Some writers on probability call this the "conditional covariance formula"[2] or use other names.

Note: The conditional expected values E( X | Z ) and E( Y | Z ) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is g(Z). Similar comments apply to the conditional covariance.


The law of total covariance can be proved using the law of total expectation: First,

[math]\displaystyle{ \operatorname{cov}(X,Y) = \operatorname{E}[XY] - \operatorname{E}[X]\operatorname{E}[Y] }[/math]

from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z:

[math]\displaystyle{ = \operatorname{E}\big[\operatorname{E}[XY\mid Z]\big] - \operatorname{E}\big[\operatorname{E}[X\mid Z]\big]\operatorname{E}\big[\operatorname{E}[Y\mid Z]\big] }[/math]

Now we rewrite the term inside the first expectation using the definition of covariance:

[math]\displaystyle{ = \operatorname{E}\!\big[\operatorname{cov}(X,Y\mid Z) + \operatorname{E}[X\mid Z]\operatorname{E}[Y\mid Z]\big] - \operatorname{E}\big[\operatorname{E}[X\mid Z]\big]\operatorname{E}\big[\operatorname{E}[Y\mid Z]\big] }[/math]

Since expectation of a sum is the sum of expectations, we can regroup the terms:

[math]\displaystyle{ = \operatorname{E}\!\left[\operatorname{cov}(X,Y\mid Z)] + \operatorname{E}[\operatorname{E}[X\mid Z] \operatorname{E}[Y\mid Z]\right] - \operatorname{E}[\operatorname{E}[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]] }[/math]

Finally, we recognize the final two terms as the covariance of the conditional expectations E[X | Z] and E[Y | Z]:

[math]\displaystyle{ = \operatorname{E}\big[\operatorname{cov}(X,Y \mid Z)\big]+\operatorname{cov}\big(\operatorname{E}[X\mid Z],\operatorname{E}[Y\mid Z]\big) }[/math]

See also

Notes and references

  1. Matthew R. Rudary, On Predictive Linear Gaussian Models, ProQuest, 2009, page 121.
  2. Sheldon M. Ross, A First Course in Probability, sixth edition, Prentice Hall, 2002, page 392.