WaldsEquation 2004-10-01 16:57:18 2006-09-16 11:20:36 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $X_1, X_2,\ldots, X_N$ be a sequence of $N$ iid random variables distributed as random variable $X$, such that \begin{enumerate} \item $N>0$ is itself a random variable (integer-valued), \item the expectation of $X$, $\operatorname{E}[X]<\infty$, and \item $\operatorname{E}[N]<\infty$. \end{enumerate} Then $$\operatorname{E}\Big[\sum_{i=1}^{N}X_i\Big]=\operatorname{E}[N]\operatorname{E}[X].$$ \par The integer $N$ from above can be viewed as a stopping time for the stochastic process $\lbrace X_i \mid i\in\mathbb{Z}^+ \rbrace$.