Wald’s equation

Let $X_{1},X_{2},\ldots,X_{N}$ be a sequence of $N$ iid random variables distributed as random variable $X$ , such that

1.

$N>0$ is itself a random variable (integer-valued),
2.

the expectation of $X$ , $\operatorname{E}[X]<\infty$ , and
3.

$\operatorname{E}[N]<\infty$ .

Then

\operatorname{E}\Big{[}\sum_{i=1}^{N}X_{i}\Big{]}=\operatorname{E}[N]% \operatorname{E}[X].

The integer $N$ from above can be viewed as a stopping time for the stochastic process $\{X_{i}\mid i\in\mathbb{Z}^{+}\}$ .

Title	Wald’s equation
Canonical name	WaldsEquation
Date of creation	2013-03-22 14:40:04
Last modified on	2013-03-22 14:40:04
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 60K05
Classification	msc 60G40