Generalizations of Distributions Related to ($k_1,k_2$)-runs

TL;DR

This paper explores generalized distributions related to ($k_1,k_2$)-runs in Bernoulli sequences, deriving their properties and demonstrating applications.

Contribution

It introduces new generalized distributions for ($k_1,k_2$)-runs and provides explicit formulas for their probability functions and moments.

Findings

Derived probability generating functions and mass functions.

Analyzed distributional properties and moments.

Presented applications demonstrating relevance.

Abstract

The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, two applications to demonstrate the relevance of the results are given.