Blackwell-Girshick equation

The Blackwell-Girshick equation is an equation in stochastics that can be used to calculate the variance of random sums of random variables .

It is named after David Blackwell and Abe Girshick .

statement

If a random variable with values ​​in and are independent and identically distributed random variables, which are also independent of, and exists for all and the second moment , then the by has ${\displaystyle N}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle (X_{i})_{i\in \mathbb {N} _{0}}}$ ${\displaystyle N}$ ${\displaystyle X_{i}}$ ${\displaystyle N}$

${\displaystyle Y:=\sum _{i=1}^{N}X_{i}}$

defined random variable the variance

${\displaystyle \operatorname {Var} (Y)=\operatorname {Var} (N)(\operatorname {E} (X_{0}))^{2}+\operatorname {E} (N)\operatorname {Var} (X_{0})}$.

The Blackwell-Girshick equation can be derived using the conditional variance and the variance decomposition . If these are also random variables , the derivation can be done in an elementary way using the chain rule and the probability-generating function . ${\displaystyle X_{i}}$ ${\displaystyle \mathbb {N} }$

example

Let Poisson distributed to the expected value and the Bernoulli distributed to the parameter . Then ${\displaystyle N}$ ${\displaystyle \lambda }$ ${\displaystyle X_{i}}$ ${\displaystyle p}$

${\displaystyle \operatorname {Var} (Y)=\lambda p^{2}+\lambda p(1-p)=\lambda p}$.

Usage and related concepts

The Blackwell-Girshick equation is used in non-life insurance to calculate the variance of composite distributions, such as the Poisson composite distribution . Wald's formula provides similar statements about the expected value of composite distributions .

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .