# 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. 1.

$f_{X}(\alpha)\neq\operatorname{min}(f_{X}(x))$,

2. 2.

$f_{X}(\alpha)\geq f_{X}(z)$ 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 $\Omega=\{0,1,2,2,3,4,4,4,5,5,6,7,8\}$ is the sample space for the random variable $X$, then the mode of the distribution function $f_{X}$ is 4.

• if $\Omega=\{0,2,4,5,6,6,7,9,11,11,14,18\}$ 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 $np(1-p)$, the mode is

 $\{\alpha\mid p(n+1)-1\leq\alpha\leq p(n+1)\}.$
• 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.

• 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 Mode 2013-03-22 14:23:33 2013-03-22 14:23:33 CWoo (3771) CWoo (3771) 4 CWoo (3771) Definition msc 60A99