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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:
- $f_X(\alpha)\neq \operatorname{min}(f_X(x))$ ,
- $f_X(\alpha)\geq f_X(z)$ for all $z\in \mathbb{R}$ .
The mode of $f_X$ 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 $\Omega=\lbrace 0,1,2,2,3,4,4,4,5,5,6,7,8 \rbrace$ is the sample space for the random variable $X$ , then the mode of the distribution function $f_X$ is 4.
- 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.
- For a binomial distribution with mean $np$ and variance $np(1-p)$ , the mode is $$\lbrace \alpha \mid p(n+1)-1\leq \alpha \leq p(n+1) \rbrace.$$
- 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.
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Cross-references: exponential distributions, location parameter, parameter, gamma distribution, standard deviation, normal distribution, integer, integral, Poisson distribution, variance, mean, binomial distribution, distribution function, contains, elements, real number, random variable, function, density, distribution
There are 20 references to this entry.
This is version 1 of mode, born on 2004-06-04.
Object id is 5889, canonical name is Mode.
Accessed 6405 times total.
Classification:
| AMS MSC: | 60A99 (Probability theory and stochastic processes :: Foundations of probability theory :: Miscellaneous) |
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Pending Errata and Addenda
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