NegativeBinomialRandomVariable 2001-10-26 03:59:47 2004-02-14 06:34:04 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{graphicx} %\usepackage{xypic} $X$ is a \emph{negative binomial random variable} with parameters $r$ and $p$ if\\ \par $f_X(x) ={r+x-1 \choose x} p^r (1-p)^x$, $x=\{0,1,...\}$ \\ \par Parameters:\\ \par \begin{list}{$\star$ }{} \item $r > 0$ \item $p \in [0,1]$ \end{list} \par Syntax:\\ \par $X\sim NegBin(r,p)$\\ \par Notes:\\ \par \begin{enumerate} \item If $r \in \mathbb{N}$, $X$ represents the number of failed Bernoulli trials before the $r$th success. Note that if $r=1$ the variable is a geometric random variable. \item $E[X] = r \frac{1-p}{p}$ \item $Var[X] = r \frac{1-p}{p^2}$ \item $M_X(t) = (\frac{p}{1 - (1-p)e^t})^r$ \end{enumerate}