We study the distributions of waiting times in variations of the negative binomial distribution of order $k$. One variation apply different enumeration scheme on the runs of successes. Another case considers binary trials for which the probability of ones is geometrically varying. We investigate the exact distribution of the waiting time for the $r$-th occurrence of success run of a specified length (non-overlapping, overlapping, at least, exactly, $\ell$-overlapping) in a $q$-sequence of binary trials. The main theorems are Type $1$, $2$, $3$ and $4$ $q$-negative binomial distribution of order $k$ and $q$-negative binomial distribution of order $k$ in the $\ell$-overlapping case. In the present work, we consider a sequence of independent binary zero and one trials with not necessarily identical distribution with the probability of ones varying according to a geometric rule. Exact formulae for the distributions obtained by means of enumerative combinatorics.
翻译:在二进制试验中,我们用负二进制分配法来研究等候时间的分配情况。一个变式对成功的运行情况采用不同的查点办法。另一个案件考虑了二进制试验,其可能性是几何不同的。我们调查了在美元/美元重叠案中,在一定长度(不重叠、重叠、至少准确、美元-重叠)的成功运行中,等待时间的确切分配情况。在二进制试验中,主要理论是1美元、2美元、3美元和4美元-负制的二进制试验。在美元/美元重复案件中,我们考虑了单进制试验的等待时间的准确分配情况。在目前的工作中,我们考虑的是独立的二进制试验的顺序和一种试验的顺序,其分配概率不一定与几进制规则不同。通过数字组合法获得的分布的精确公式。