#### Publication Date

2-2017

#### Abstract

If we have two random variables ξ_{1} and ξ_{1}, then we can form their *mixture* if we take ξ_{1} with some probability w and ξ_{2} 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 * ρ_{1}(x) +( 1 − w) * ρ_{2}(x). A natural question is: can we use other functions f(ρ_{1}, ρ_{2}) to combine the pdfs, i.e., to produce a new pdf ρ(x) =f(ρ_{1}(x), ρ_{2}(x))? In this paper, we prove that the only combination operations that always lead to a pdf are the operations f(ρ_{1}, ρ_{2})=w * ρ_{1} + (1 − w) * ρ_{2} corresponding to mixture.

## Comments

Technical Report: UTEP-CS-17-15

To appear in

International Journal of Intelligent Technologies and Applied Statistics IJITAS