Universal parameterized family of distributions of runs

We present explicit formulae for parameterized families of distributions of the number of nonoverlapping words and increasing nonverlapping words in independent and identically distributed (i.i.d.) finite valued random variables, respectively. Then we provide an explicit formula for a parameterized family of distributions of the number of runs, which generalizes \(\mu\)-overlapping distributions for \(\mu\geq 0\) in i.i.d.~binary valued random variables. We also demonstrate that of runs whose size are exactly given numbers (Mood 1940). The number of arithmetic operations required to compute our formula for generalized distributions of runs for fixed number of parameters and fixed range is linear order of sample size.