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 ≤ k ≤ n . Theorem 3.1 gives an closed expression for the probability mass function (PMF) of the Type IV 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 Type IV 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.