mode
Given a probability distribution (density) function ${f}_{X}(x)$ with random variable^{} $X$ and $x\in \mathbb{R}$, a mode of ${f}_{X}(x)$ is a real number $\alpha $ such that:

1.
${f}_{X}(\alpha )\ne \mathrm{min}({f}_{X}(x))$,

2.
${f}_{X}(\alpha )\ge {f}_{X}(z)$ for all $z\in \mathbb{R}$.
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 unimodal. If it has two elements, then ${f}_{X}$ is called bimodal. When ${f}_{X}$ has more than two modes, it is called multimodal.

•
if $\mathrm{\Omega}=\{0,1,2,2,3,4,4,4,5,5,6,7,8\}$ is the sample space for the random variable $X$, then the mode of the distribution function^{} ${f}_{X}$ is 4.

•
if $\mathrm{\Omega}=\{0,2,4,5,6,6,7,9,11,11,14,18\}$ is the sample space for $X$, then the modes of ${f}_{X}$ are 6 and 11 and ${f}_{X}$ is bimodal.

•
For a binomial distribution with mean $np$ and variance^{} $np(1p)$, the mode is
$$\{\alpha \mid p(n+1)1\le \alpha \le p(n+1)\}.$$ 
•
For a Poisson distribution^{} with integral sample space and mean $\lambda $, if $\lambda $ is nonintegral, 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.

•
For a normal distribution^{} with mean $\mu $ and standard deviation^{} $\sigma $, the mode is $\mu $.

•
For a gamma distribution^{} with the shape parameter $\gamma $, location parameter $\mu $, and scale parameter $\beta $, the mode is $\gamma 1$ if $\gamma >1$.

•
Both the Pareto and the exponential distributions^{} have mode = 0.
Title  mode 

Canonical name  Mode 
Date of creation  20130322 14:23:33 
Last modified on  20130322 14:23:33 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  4 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60A99 