On the modes of the negative binomial distribution of order k

ABSTRACT

In this paper, the modes of the negative binomial distribution of order k are studied. Firstly, the method of transition probability flow graphs is introduced to deal with the probability-generating function of the geometric distribution of order k, which is a special case of the negative binomial distribution of the same order. And then, the general negative binomial distribution of order k is investigated. By means of probability distribution function, the mode of the geometric distribution of order k is derived, i.e. $m_{X_{\left(k\right)}}=k$. Based on the Fibonacci sequence and Poly-nacci sequence, the modes of the negative binomial distribution of order k in some cases are obtained: (1) $m_{X_{(2,2)}}=6,7,8$ and $m_{X_{(3,2)}}=16$, for p=0.5; (2) $m_{X_{(2,3)}}=13$ for p=0.5. Finally, an application of negative binomial distribution of order k in continuous sampling plans is given.

KEYWORDS:

• Negative binomial distribution of order k
• success run
• Fibonacci sequence
• transition probability flow graphs
• probability-generating function
• mode

AMS SUBJECT CLASSIFICATION:

• 60A99
• 60J20

Acknowledgments

The authors are grateful for the useful suggestions made by the referees and Professor A.N.Philippou, which greatly improved the presentation, and are grateful to the editor for his encouragement.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The paper was supported by Fundamental Research Funds for the Central Universities (No. S11JB00400) and National Natural Science Foundation of China (No. 71271026), during which this work was carried out.