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.
be a random variable with values in
be independent and identically distributed random variables, which are also independent of
, and assume that the second moment exists for all
Then, the random variable defined by has the variance The Blackwell-Girshick equation can be derived using conditional variance and variance decomposition.
are natural number-valued random variables, the derivation can be done elementarily using the chain rule and the probability-generating function.
be the random variable which is 1 if
Then By Wald's equation, under the given hypotheses,
have a Poisson distribution with expectation
λ
follow a Bernoulli distribution with parameter
is also Poisson distributed with expectation
λ p
λ p
We can check this with the Blackwell-Girshick equation:
λ
, so we must have 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.