Formula of forest

In stochastics, the formula of Wald or Wald's identity is an equation with the help of which the expected value of sums of random variables with a random number of summands can be calculated. It was proven in 1944 in a work by the mathematician Abraham Wald .

formulation

Let it be a sequence of independent, identically distributed , integrable random variables and a -value random variable with , which is independent of the sequence . Then applies ${\displaystyle (X_{n})_{n\geq 1}}$ ${\displaystyle T}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \operatorname {E} (T)<\infty }$ ${\displaystyle (X_{n})}$

${\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} (T)\operatorname {E} (X_{1})}$.

proof

Because is independent of the sequence , it follows from conditions on the value of : ${\displaystyle T}$ ${\displaystyle (X_{n})}$ ${\displaystyle T}$

${\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T=n\right)=\operatorname {E} \left(\sum _{k=1}^{n}X_{k}\right)=\sum _{k=1}^{n}\operatorname {E} (X_{k})=n\operatorname {E} (X_{1})}$,

so

${\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T\right)=T\operatorname {E} (X_{1})}$.

By applying the expectation to this equation, one finally obtains

${\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} \left(\operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T\right)\right)=\operatorname {E} (T\operatorname {E} (X_{1}))=\operatorname {E} (T)\operatorname {E} (X_{1})}$.

If they are all valued, the elementary proof can also take place via probability-generating functions using the chain rule . ${\displaystyle X_{i}}$ ${\displaystyle \mathbb {N} _{0}}$

Generalization to stop times

Let it now be a sequence of identically distributed, integrable random variables that is adapted to a filtering process , i.e. it is measurable for all . If by independently for all and an integrable stop time with respect , so also the formula is of forest: ${\displaystyle (X_{n})_{n\geq 1}}$ ${\displaystyle ({\mathcal {F}}_{n})_{n}}$ ${\displaystyle n}$ ${\displaystyle X_{n}}$ ${\displaystyle {\mathcal {F}}_{n}}$ ${\displaystyle X_{n+1}}$ ${\displaystyle {\mathcal {F}}_{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle T}$ ${\displaystyle ({\mathcal {F}}_{n})_{n}}$

${\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} (T)\operatorname {E} (X_{1})}$.

Related concepts

Similar statements about the variance of composite distributions can be made with the Blackwell-Girshick equation .

Individual evidence

1. ↑ Abraham Wald: On Cumulative Sums of Random Variables. In: The Annals of Mathematical Statistics No. 15, Vol. 3, pp. 283-296, doi : 10.1214 / aoms / 1177731235 .
2. ↑ David Meintrup, Stefan Schäffler: Stochastics. Theory and applications. Springer, Berlin / Heidelberg 2005, ISBN 3-540-21676-6 , p. 287.
3. ↑ Heinz Bauer : Probability Theory. 5th edition. De Gruyter textbook, Berlin 2002, ISBN 3-11-017236-4 , chapter 17.