PlanetMath: mode

(more info)

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mode

(Definition)

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

$f_X(\alpha)\neq \operatorname{min}(f_X(x))$ ,
$f_X(\alpha)\geq f_X(z)$ for all $z\in \mathbb{R}$ .

The mode of is the set of all modes of (It is also customary to say denote the mode of to be elements within the mode of ). If the mode contains one element, then we say that is unimodal. If it has two elements, then is called bimodal. When has more than two modes, it is called multimodal.

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 , then the mode of the distribution function is 4.
if $\Omega=\lbrace 0,2,4,5,6,6,7,9,11,11,14,18 \rbrace$ is the sample space for , then the modes of are 6 and 11 and is bimodal.
For a binomial distribution with mean and variance , the mode is
$\displaystyle \lbrace \alpha \mid p(n+1)-1\leq \alpha \leq p(n+1) \rbrace.$
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.