Wald’s equation

Let $X_1,X_2,\mathrm{\dots },X_N$ 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 $X$, , and

3. .

Then

$$\mathrm{E}\left[\sum _{i=1}^N{X}_i\right]=\mathrm{E}[N]\mathrm{E}[X].$$

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