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mode


Given a probability distribution (density) function fX(x) with random variableMathworldPlanetmath X and x, a mode of fX(x) is a real number α such that:

  1. 1.

    fX(α)min(fX(x)),

  2. 2.

    fX(α)fX(z) for all z.

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

  • if Ω={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 functionMathworldPlanetmath fX is 4.

  • if Ω={0,2,4,5,6,6,7,9,11,11,14,18} is the sample space for X, then the modes of fX are 6 and 11 and fX is bimodal.

  • For a binomial distribution with mean np and varianceMathworldPlanetmath np(1-p), the mode is

    {αp(n+1)-1αp(n+1)}.
  • For a Poisson distributionMathworldPlanetmath with integral sample space and mean λ, if λ is non-integral, then the mode is the largest integer less than or equal to λ; if λ is an integer, then both λ and λ-1 are modes.

  • For a normal distributionMathworldPlanetmath with mean μ and standard deviationMathworldPlanetmath σ, the mode is μ.

  • For a gamma distributionMathworldPlanetmath with the shape parameter γ, location parameter μ, and scale parameter β, the mode is γ-1 if γ>1.

  • Both the Pareto and the exponential distributionsMathworldPlanetmath have mode = 0.

Title mode
Canonical name Mode
Date of creation 2013-03-22 14:23:33
Last modified on 2013-03-22 14:23:33
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Definition
Classification msc 60A99