On fuzzy random variables: examples and generalizations

Year, pages: 2005, 175 - 185

There are random experiments in which the notion of a classical random variable, as a map sending each elementary event to a real number, does not capture their nature. This leads to fuzzy random variables in the Bugajski–Gudder sense. The idea is to admit variables sending the set $Ω$ of elementary events not into the real numbers, but into the set $M₁⁺(Bbb R)$ of all probability measures on the real Borel sets (each real number $rin Bbb R$ is considered as the degenerated probability measure $δ_r$ concentrated at $r$). We start with four examples of random experiments ($Ω$ is finite); the last one is more complex, it generalizes the previous three, and it leads to a general model. A fuzzy random variable is a map $φ$ of $M₁⁺(Ω)$ into $M₁⁺(Ξ)$, where $M₁⁺(Ξ)$ is the set of all probability measures on another measurable space $(Ξ,B(Ξ))$, satisfying certain measurability condition. We show that for discrete spaces the measurability condition holds true. We continue in our effort to develop a suitable theory of $ID$-posets, $ID$-random variables, and $ID$-observables. Fuzzy random variables and Markov kernels become special cases.

Papčo, M. 2005. On fuzzy random variables: examples and generalizations. In Tatra Mountains Mathematical Publications, vol. 30, no.1, pp. 175-185. 1210-3195.

Papčo, M. (2005). On fuzzy random variables: examples and generalizations. Tatra Mountains Mathematical Publications, 30(1), 175-185. 1210-3195.