Abstract
The conditional mean value has applications in regression analysis and in financial mathematics, because they are used in it. We can find papers from recent years that use the conditional mean value in fuzzy cases. As the intuitionstic fuzzy sets are an extension of fuzzy sets, we will try to define a conditional mean value for the intuitionistic fuzzy case. The conditional mean value in crisp intuitionistic fuzzy events was first studied by V. Valenčáková in 2009. She used Gödel connectives. Her approach can only be used for special cases of intuitionistic fuzzy events, therefore, we want to define a conditional mean value for all elements of a family of intuitionistic fuzzy events. In this paper, we define the conditional mean value for intuitionistic fuzzy events using Lukasiewicz connectives. We use a Kolmogorov approach and the notions from a classical probability theory for construction. B. Riečan formulated a conditional intuitionistic fuzzy probability for intuitionistic fuzzy events using an intuitionistic fuzzy state in 2012. In classical cases, there exists a connection between the conditional probability and the conditional mean value, therefore we show a connection between the conditional intuitionistic fuzzy probability induced by the intuitionistic fuzzy state and the conditional intuitionistic fuzzy mean value.
Highlights
A conditional mean value has many applications in regression analysis and in financial mathematics and insurance
As there are known practical applications in classical cases and fuzzy cases, it is interesting to study a notion of conditional expectation in a family of intuitionistic fuzzy sets
This paper is concerned with the probability theory of intuitionistic fuzzy sets
Summary
A conditional mean value has many applications in regression analysis and in financial mathematics and insurance. Jurecková studied a notion of conditional expectation of observables on MV-algebras of fuzzy sets and on probability MV-algebras with product (see [6,7]). Valencáková defined a conditional mean value E(x | y) for crisp intuitionistic fuzzy events A = {(χA, 1 − χA)} ⊂ F = {(μA, νA) ; μA + νA ≤ 1Ω} as a Borel function g : R → R satisfying the following equality. We define a conditional mean value for the family of intuitionistic fuzzy events F. The paper is organized as follows: Section 2 includes the basic notions from intuitionistic fuzzy probability theory as an intuitionistic fuzzy event, an intuitionistic fuzzy state, an intuitionistic fuzzy observable and an intuitionistic fuzzy mean value. We define a conditional intuitionistic fuzzy mean value for the intuitionistic fuzzy events.
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