Blackwell-Girshick equation

The Blackwell-Girshick equation is an equation in probability theory that allows for the calculation of the variance of random sums of random variables.[1] It is the equivalent of Wald's lemma for the expectation of composite distributions.

It is named after David Blackwell and Meyer Abraham Girshick.

Statement

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Let   be a random variable with values in  , let   be independent and identically distributed random variables, which are also independent of  , and assume that the second moment exists for all   and  . Then, the random variable defined by

 

has the variance

 .

The Blackwell-Girshick equation can be derived using conditional variance and variance decomposition. If the   are natural number-valued random variables, the derivation can be done elementarily using the chain rule and the probability-generating function.[2]

Proof

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For each  , let   be the random variable which is 1 if   equals   and 0 otherwise, and let  . Then

 

By Wald's equation, under the given hypotheses,  . Therefore,

 

as desired.[3]: §5.1, Theorem 5.10 

Example

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Let   have a Poisson distribution with expectation  , and let   follow a Bernoulli distribution with parameter  . In this case,   is also Poisson distributed with expectation  , so its variance must be  . We can check this with the Blackwell-Girshick equation:   has variance   while each   has mean   and variance  , so we must have

 .
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The Blackwell-Girshick equation is used in actuarial mathematics to calculate the variance of composite distributions, such as the compound Poisson distribution. Wald's equation provides similar statements about the expectation of composite distributions.

Literature

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  • For an example of an application: Mühlenthaler, M.; Raß, A.; Schmitt, M.; Wanka, R. (2021). "Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on sorting and OneMax". Natural Computing: 1–27.

References

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  1. ^ Blackwell, D. A.; Girshick, M. A. (1979). Theory of games and statistical decisions. Courier Corporation.
  2. ^ Achim Klenke (2013), Wahrscheinlichkeitstheorie (3rd ed.), Berlin Heidelberg: Springer-Verlag, p. 109, doi:10.1007/978-3-642-36018-3, ISBN 978-3-642-36017-6, S2CID 242882110
  3. ^ Probability Theory : A Comprehensive Course, Achim Klenke, London, Heidelberg, New York, Dordrecht: Springer, 2nd ed., 2014, ISBN 978-1-4471-5360-3.