Why Mixture of Probability Distributions

Authors

Andrzej Pownuk, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

2-2017

Comments

Technical Report: UTEP-CS-17-15

To appear in International Journal of Intelligent Technologies and Applied Statistics IJITAS

Abstract

If we have two random variables ξ₁ and ξ₁, then we can form their mixture if we take ξ₁ with some probability w and ξ₂ with the remaining probability 1 − w. The probability density function (pdf) ρ(x) of the mixture is a convex combination of the pdfs of the original variables: ρ(x) = w * ρ₁(x) +( 1 − w) * ρ₂(x). A natural question is: can we use other functions f(ρ₁, ρ₂) to combine the pdfs, i.e., to produce a new pdf ρ(x) =f(ρ₁(x), ρ₂(x))? In this paper, we prove that the only combination operations that always lead to a pdf are the operations f(ρ₁, ρ₂)=w * ρ₁ + (1 − w) * ρ₂ corresponding to mixture.