AUTHORS: Jiguang Shao, Qiuyuan Wang

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ABSTRACT: This paper introduces three generalized geometric distributions: the binomial, negative binomial and Poisson distribution of the same order k. The generating functions and probability distributions of them are investigated, and then the corresponding modes of the distributions are discussed. By the Fibonacci sequence, the modes of the negative binomial distribution of order k are derived as mX(2,2) = 6, 7, 8 and mX(2,3) = 13. For the mode of the binomial distribution of order k, only a conjecture is proposed as an open question for the parameters k = 2, n = 2ň and p = 0.5. Finally, the modes of the Poisson distribution of order k are discussed in some cases.

KEYWORDS: mode, success run, probability generating function, negative binomial distribution of order k, Poisson distribution of order k, binomial distribution of order k

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