On distribution of runs and patterns in four state trials

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 $X_1,X_2,\dots$ be a sequence of four state i.i.d.\ trials taking values in the set $\mathcal{S}=\{A,B,C,D\}$ 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.