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.


  3. 3.

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



  • •


  • •

    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:


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


    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):


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


    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