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.

翻译：我们分别给出了独立同分布（i.i.d.）有限值随机变量中非重叠词和递增非重叠词数量的参数化分布族的显式公式。随后，我们提供了游程数量参数化分布族的显式公式，该公式推广了独立同分布二值随机变量中 \(\mu\geq 0\) 的 \(\mu\)-重叠分布。同时，我们还展示了游程大小恰好为给定数字的情况（Mood 1940）。对于固定参数数量和固定范围，计算我们提出的广义游程分布公式所需的算术运算次数与样本量呈线性关系。