The study of discrete probabilistic distributions of random sets of events using associative function

Natalia A. Lukyanova ; Daria V. Semenova

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 4, pp. 500-514

Résumé

In this work the class of discrete probabilistic distributions of the II-nd type of random sets of event is investigated. As the tool for constructing of such probabilistic distributions it is offered to use associative functions. There is stated a new approach to define a discrete probabilistic distribution of the II-nd type of a random set on a finite set of $N$ events on the basis of obtained recurrence relation and a given associative function. Advantage of the offered approach is that for definition of probabilistic distribution instead of a totality of $2^N$ probabilities it is enough to know $N$ probabilities of events and a type of associative function. In this paper an $|X|$-ary covariance of a random set of events is considered. It is a measure of the additive deviation of the events from the independent situation. The process of recurrent constructing a probabilistic distribution II-nd type is demonstrated by the example of three associative functions. The proof of the legitimacy / illegitimacy the obtained distribution by passing to the probabilistic distribution of the I-st type by formulas of Möbius is given. Theorems that establish the form and conditions of the legitimacy of the resulting probabilistic distributions are proved. $|X|$-ary covariances of random sets of events are found.

Détail

Keywords: random set of events, discrete probabilistic distributions, associative function, $|X|$-ary covariance.