  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.
