Discrete distributions related to success runs of lengthk in aulti-state markov chain

Abstract

We consider the time-homogeneous multi-state Markov chain {X , n ≥0} with states labeled as “0” (success) and “f” (failure), f = 1,2,… Let k be a fixed positive integer and SK be the event of a success run of length k in the sequence X⁰: X ,… In this article, joint probability generating functions for various statistics, related to the event SK₀ are derived. In particular cases the exact distribution of the total number of successes S, the total number of failures F and the total number of trials N are deduced. The distribution of N is called the geometric distribution of order k for time-homogeneous {0,1,2,…}-valued Markov chain.

Keywords:

• Geometric distribution of orderk
• Joint probability generating function
• Time homogeneous {0,l,…}-valued Markov chain
• Random walk
• Recursion
• Urn model