multinomial distribution


Let 𝐗=(X1,…,Xn) be a random vector such that

  1. 1.

    Xi≥0 and Xi∈ℤ

  2. 2.

    X1+⋯+Xn=N, where N is a fixed integer

Then X has a multinomial distributionMathworldPlanetmath if there exists a parameter vector 𝝅=(π1,…,πn) such that

  1. 1.

    πi≥0 and πi∈ℝ

  2. 2.

    π1+⋯+πn=1

  3. 3.

    X has a discrete probability distribution function f𝐗⁢(𝒙) in the form:

    f𝐗⁢(𝒙)=N!x1!⁢⋯⁢xn!⁢∏i=1nπixi

Remarks

  • •

    E⁡[𝐗]=N⁢𝝅

  • •

    Var⁡[𝐗]=(vi⁢j), where

    vi⁢j={N⁢πi⁢(1-πi)if i=j;-N⁢πi⁢πjif i≠j.
  • •

    When n=2, the multinomial distribution is the same as the binomial distribution

  • •

    If X1,…,Xn are mutually independent Poisson random variables with parameters λ1,…,λn respectively, then the conditionalMathworldPlanetmathPlanetmath joint distributionPlanetmathPlanetmath of X1,…,Xn given that X1+⋯+Xn=N is multinomial with parameters λi/λ, where λ=∑λi.

    Sketch of proof. Each Xi is distributed as:

    fXi⁢(xi)=e-λi⁢λixixi!

    The mutual independence of the Xi’s shows that the joint probability distribution of the Xi’s is given by

    f𝐗⁢(𝒙)=∏i=1ne-λi⁢λixixi!=e-λ⁢∏i=1nλixixi!,

    where 𝐗=(X1,…,Xn), 𝒙=(x1,…,xn) and λ=λ1+⋯+λn. Next, let X=X1+⋯+Xn. Then X is Poisson distributed with parameter λ (which can be shown by using inductionMathworldPlanetmath and the mutual independence of the Xi’s):

    fX⁢(x)=e-λ⁢λxx!.

    The conditional probability distribution of X given that X=N is thus given by:

    f𝐗(𝒙∣X=N)=f𝐗⁢(𝒙)fX⁢(N)=(e-λ∏i=1nλixixi!)/(e-λ⁢λNN!)=N!x1!⁢⋯⁢xn!∏i=1n(λiλ)xi,

    where ∑xi=N and that ∑λi/λ=1.

Title multinomial distribution
Canonical name MultinomialDistribution
Date of creation 2013-03-22 14:33:35
Last modified on 2013-03-22 14:33:35
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 60E05