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
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{\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)).}
The nomenclature in this article's title parallels the phrase law of total variance conditional covariance formula"[ 2]
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,
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{\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}
from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z :
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{\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 ]}}
Now we rewrite the term inside the first expectation using the definition of covariance:
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{\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 ]}}
Since expectation of a sum is the sum of expectations, we can regroup the terms:
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{\displaystyle =\operatorname {E} \!{\big [}\operatorname {cov} (X,Y\mid Z){\big ]}+\operatorname {E} {\big [}\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 ]}}
Finally, we recognize the final two terms as the covariance of the conditional expectations E[X | Z ] and E[Y | Z ]:
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{\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 )}}
Notes and references
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^ Matthew R. Rudary, On Predictive Linear Gaussian Models , ProQuest, 2009, page 121.
^ Sheldon M. Ross, A First Course in Probability , sixth edition, Prentice Hall, 2002, page 392.
![{\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1640f54ab44b3f8b7b3fd6ce9b44e47f6576700)
![{\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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e73873e9bb5e80b4f254985371019560dde3459e)
![{\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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88000aa872d3c4aec4ed610180f7d8d2a2329308)
![{\displaystyle =\operatorname {E} \!{\big [}\operatorname {cov} (X,Y\mid Z){\big ]}+\operatorname {E} {\big [}\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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1db8ce13ab9fb61360993c72d8d16df4f304b33)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3010c7ce77176b93bc8b3cbf13c7c9b789843c)