Mode 2004-06-04 16:46:01 2004-06-04 16:46:01 Definition % 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 Given a probability distribution (density) function $f_X(x)$ with random variable $X$ and $x\in \mathbb{R}$, a \emph{mode} of $f_X(x)$ is a real number $\alpha$ such that: \begin{enumerate} \item $f_X(\alpha)\neq \operatorname{min}(f_X(x))$, \item $f_X(\alpha)\geq f_X(z)$ for all $z\in \mathbb{R}$. \end{enumerate} \emph{The} mode of $f_X$ is the set of all modes of $f_X$ (It is also customary to say denote the mode of $f_X$ to be elements within the mode of $f_X$). If the mode contains one element, then we say that $f_X$ is \emph{unimodal}. If it has two elements, then $f_X$ is called \emph{bimodal}. When $f_X$ has more than two modes, it is called \emph{multimodal}. \begin{itemize} \item if $\Omega=\lbrace 0,1,2,2,3,4,4,4,5,5,6,7,8 \rbrace$ is the sample space for the random variable $X$, then the mode of the distribution function $f_X$ is 4. \item if $\Omega=\lbrace 0,2,4,5,6,6,7,9,11,11,14,18 \rbrace$ is the sample space for $X$, then the modes of $f_X$ are 6 and 11 and $f_X$ is bimodal. \item For a binomial distribution with mean $np$ and variance $np(1-p)$, the mode is $$\lbrace \alpha \mid p(n+1)-1\leq \alpha \leq p(n+1) \rbrace.$$ \item For a Poisson distribution with integral sample space and mean $\lambda$, if $\lambda$ is non-integral, then the mode is the largest integer less than or equal to $\lambda$; if $\lambda$ is an integer, then both $\lambda$ and $\lambda-1$ are modes. \item For a normal distribution with mean $\mu$ and standard deviation $\sigma$, the mode is $\mu$. \item For a gamma distribution with the shape parameter $\gamma$, location parameter $\mu$, and scale parameter $\beta$, the mode is $\gamma-1$ if $\gamma>1$. \item Both the Pareto and the exponential distributions have mode = 0. \end{itemize}