# 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. 1.

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

2. 2.

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

3. 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 WaldsEquation 2013-03-22 14:40:04 2013-03-22 14:40:04 CWoo (3771) CWoo (3771) 8 CWoo (3771) Theorem msc 60K05 msc 60G40