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.