mode
Given a probability distribution (density) function fX(x) with random variable X and x∈ℝ, a mode of fX(x) is a real number α such that:
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1.
fX(α)≠min(fX(x)),
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2.
fX(α)≥fX(z) for all z∈ℝ.
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.
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•
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 function
fX is 4.
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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.
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•
For a binomial distribution with mean np and variance
np(1-p), the mode is
{α∣p(n+1)-1≤α≤p(n+1)}. -
•
For a Poisson distribution
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.
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For a normal distribution
with mean μ and standard deviation
σ, the mode is μ.
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•
For a gamma distribution
with the shape parameter γ, location parameter μ, and scale parameter β, the mode is γ-1 if γ>1.
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•
Both the Pareto and the exponential distributions
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 |