Order isomorphism: Difference between revisions
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In [[probability theory]], the '''law of total covariance''',<ref>Matthew R. Rudary, ''On Predictive Linear Gaussian Models'', ProQuest, 2009, page 121.</ref> '''covariance decomposition formula''', or '''ECCE''' states that if ''X'', ''Y'', and ''Z'' are [[random variable]]s on the same [[probability space]], and the [[covariance]] of ''X'' and ''Y'' is finite, then | |||
:<math>\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"<ref>Sheldon M. Ross, ''A First Course in Probability'', sixth edition, Prentice Hall, 2002, page 392.</ref> or use other names. | |||
(The [[conditional expected value]]s E( ''X'' | ''Z'' ) and E( ''Y'' | ''Z'' ) are random variables in their own right, whose values depends on the value of ''Z''. Notice that the conditional expected value of ''X'' given the ''event'' ''Z'' = ''z'' is a function of ''z'' (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( ''X'' | ''Z'' = ''z'') = ''g''(''z'') then the random variable E( ''X'' | ''Z'' ) is just ''g''(''Z''). Similar comments apply to the conditional covariance.) | |||
==Proof== | |||
The law of total covariance can be proved using the [[law of total expectation]]: First, | |||
:<math>\operatorname{cov}[X,Y] = \operatorname{E}[XY] - \operatorname{E}[X]\operatorname{E}[Y]</math> | |||
from the definition of covariance. Then we apply the law of total expectation by conditioning on the random variable ''Z'': | |||
::<math>= \operatorname{E}[\operatorname{E}[XY\mid Z]] - \operatorname{E}[\operatorname{E}[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]]</math> | |||
Now we rewrite the term inside the first expectation using the definition of covariance: | |||
::<math>= \operatorname{E}\!\left[\operatorname{cov}[X,Y\mid Z] + \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> | |||
Since expectation of a sum is the sum of expectations, we can regroup the terms: | |||
::<math>= \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>= \operatorname{E}(\operatorname{cov}(X,Y \mid Z))+\operatorname{cov}(\operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z))</math> | |||
== Notes and references == | |||
{{reflist}} | |||
==See also== | |||
*[[Law of total variance]], a special case corresponding to ''X'' = ''Y''. | |||
==External links== | |||
{{DEFAULTSORT:Law Of Total Covariance}} | |||
[[Category:Algebra of random variables]] | |||
[[Category:Covariance and correlation]] | |||
[[Category:Articles containing proofs]] | |||
[[Category:Theory of probability distributions]] | |||
[[Category:Statistical theorems]] | |||
[[Category:Statistical laws]] | |||
Latest revision as of 15:52, 1 February 2014
In probability theory, the law of total covariance,[1] covariance decomposition formula, or ECCE 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
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.
(The conditional expected values E( X | Z ) and E( Y | Z ) are random variables in their own right, whose values depends on the value of Z. Notice that the conditional expected value of X given the event Z = z is a function of z (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is just g(Z). Similar comments apply to the conditional covariance.)
Proof
The law of total covariance can be proved using the law of total expectation: First,
from the definition of covariance. Then we apply the law of total expectation by conditioning on the random variable Z:
Now we rewrite the term inside the first expectation using the definition of covariance:
Since expectation of a sum is the sum of expectations, we can regroup the terms:
Finally, we recognize the final two terms as the covariance of the conditional expectations E[X|Z] and E[Y|Z]:
Notes and references
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See also
- Law of total variance, a special case corresponding to X = Y.