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 $X_{1}, X_{2}, \ldots$ 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.