Publication Date



Technical Report: UTEP-CS-17-15

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


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.