On calculations of exact distributions of waiting times of discrete patterns based on generating functions

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  • Statistical Science and Related Topics : Proceedings of Annual Workshop on Statistical Science and Related Topics, December 7, 2014, JOSAI UNIVERSITY

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The rational generating fuｎctions of the probabilities and　cumulative probabilities of the geometric distribution of order k are investigated. All the roots of each denominator are rigorously proved to be simple if p k/(k + 1). It is also shown that the distribution of the waiting time for (k1, k2)-events has the similar property. The results are applied to numerical calculations of probability and cumulative probability of the distributions. Further, explicit expressions for the probability functions of the geometric distributions of order 2,3, and 4 are given by using partial fraction expansion of the generating functions. Moreover, as an example that the denominator of a rational generating function has rnultiple roots. the negative binomial distribution of order 2 with parameter (r, p) are studied. Explicit expressions of the probability function and the cumulative distribution function are provided. They are also useful for numerical calculations.

Statistical Science and Related Topics : Prpceedings of Annual Workshop on Statistical Science and Reated Topics, held at Josai University on December 7 in 2014 / edited by Masatoshi IIDA, Manabu INUMA, Kiyoko NISHIZAWA, Takahiro TSUCHIDA

identifier:JOS-13447777-0805

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