- Let $\left\{x_{n}\,,\ n\geq 1\right\}$ be a sequence of real numbers.
- $N$ be a nonnegative integer valued random variable with finite expectation.
- Assume that $\,\mathrm{E}\left(S_{N}\right) = a\ \left(~\mbox{const}~\right)$ where $S_{N} = x_{1} + x_{2} + \cdots + x_{N}$.
How can we make conclusion about $N$ or $\,\mathrm{E}\left(N\right)$. Thank you !.
You cannot infer anything, for if
(1) all $x_i = 0$, and $N$ is always 1, you get $a = 0$.
(2) The $x_i$ alternate between $-1$ and $1$, and $N$ is always 2, you get $a = 0$ as well
In these two cases which give you identical data ($a$) you get different expectations for $N$ and $E(N)$.
Even if you know the $x_i$, you can't infer anything. For if all $x_i$ are $0$, for instance, you can't tell whether $N = 1$ constantly, $N = 2$ constantly, or $N$ is just some rv with finite expectation, all three of which will produce the same value of $a$.