GeometricRandomVariable 2001-10-26 03:42:16 2007-06-24 00:53:19 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A \textbf{geometric random variable} with parameter $p\in(0,1]$ is one whose density distribution function is given by \begin{equation*} f_X(x) = p(1-p)^x,\qquad x=0,1,2,\dotsc \end{equation*} This is denoted by $X\sim Geo(p)$. Notes: \begin{enumerate} \item A standard application of geometric random variables is where $X$ represents the number of failed Bernoulli trials before the first success. \item The expected value of a geometric random variable is given by $E[X] = \frac{1-p}{p}$, and the variance by $Var[X] = \frac{1-p}{p^2}$ \item The moment generating function of a geometric random variable is given by $M_X(t) = \frac{p}{1 - (1-p)e^t}$. \end{enumerate}