On Success runs of a fixed length defined on a $q$-sequence of binary trials

TL;DR

This paper derives exact and recursive distributions for success runs of fixed length in a sequence of binary trials with geometrically varying probability of ones, providing new formulas and parameter estimation methods.

Contribution

It introduces closed-form and recursive formulas for the distribution of success runs in a non-identically distributed binary sequence with geometric probability variation.

Findings

Derived closed-form PMF for the distribution of runs.

Provided recursive formulas for the distribution.

Addressed parameter estimation via numerical techniques.

Abstract

We study the exact distributions of runs of a fixed length in variation which considers binary trials for which the probability of ones is geometrically varying. The random variable $E_{n,k}$ denote the number of success runs of a fixed length $k$, $1\leq k \leq n$. Theorem 3.1 gives an closed expression for the probability mass function (PMF) of the Type4 $q$-binomial distribution of order $k$. Theorem 3.2 and Corollary 3.1 gives an recursive expression for the probability mass function (PMF) of the Type4 $q$-binomial distribution of order $k$. The probability generating function and moments of random variable $E_{n,k}$ are obtained as a recursive expression. We address the parameter estimation in the distribution of $E_{n,k}$ by numerical techniques. In the present work, we consider a sequence of independent binary zero and one trials with not necessarily identical distribution with the probability of ones varying according to a geometric rule. Exact and recursive formulae for the distribution obtained by means of enumerative combinatorics.