Run and Frequency quotas for q-binary trials

TL;DR

This paper investigates the distributions of waiting times and longest runs in binary trials with varying success probabilities and imposed quotas, providing exact formulas through combinatorial methods.

Contribution

It introduces new models for binary trials with length and frequency quotas and geometrically varying success probabilities, deriving exact distribution formulas.

Findings

Derived exact distributions for waiting times under quotas.

Analyzed longest run distributions with imposed constraints.

Extended classical models to include variable success probabilities.

Abstract

We study the distributions of waiting times in variations of the $q$-sooner and later waiting time problem. One variation imposes length and frequency quotas on the runs of successes and failures. Another case considers binary trials for which the probability of ones is geometrically varying. We also study the distributions of Longest run under the same variations. The main theorems are sooner and later waiting time problem and the joint distribution of the length of the longest success and longest failure runs when a run and frequency quotas imposed on runs of successes and failure. In the present work, we consider a sequence of independent binary $(0-1)$ trials with not necessarily identical distributed with probability of ones varying according to a geometric rule. Exact formulae for the distributions obtained by means of enumerative combinatorics.