On Success runs of a fixed length defined on a \(q\)-sequence of binary trials

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.