On distribution of runs and patterns in four state trials

TL;DR

This paper derives exact probability distributions for runs and patterns in sequences of four-state trials, with applications to DNA sequence analysis and statistical modeling of run lengths.

Contribution

It provides new exact formulas for run length distributions, longest and shortest run statistics, and waiting times in four-state sequences, advancing statistical understanding of multi-state trials.

Findings

Exact distribution formulas for runs of B's

Distribution of longest and shortest runs

Waiting time distribution for pattern occurrences

Abstract

From a mathematical and statistical point of view, a segment of a DNA strand can be viewed as a sequence of four-state (A, C, G, T) trials. We consider distributions of runs and patterns related to run lengths of multi-state sequences, especially for four states (A, B, C, D). Let $X_{1}, X_{2}, \ldots$ be a sequence of four state i.i.d.\ trials taking values in the set $\mathscr{S}=\{A,\ B,\ C,\ D\}$ of four symbols with probability $P(A)=P_{a}$, $P(B)=P_{b}$, $P(C)=P_{c}$ and $P(D)=P_{d},$ respectively. In this paper, we obtain exact formulae for the probability distribution function for runs of B's the discrete distribution of order $k$, longest run statistics, shortest run statistics, waiting time distribution and the distribution of run lengths.