Let F q X] denote a polynomial ring over a nite eld F q with q elements. Let P n be the set of monic polynomials over F q of degree n. Assuming that each of the q n possible monic polynomials in P n is equally likely, we give a complete characterization of the limiting behavior of P((n = m) as n ! 1 by a uniform asymptotic formula valid for m 1 and n?m ! 1, where n represents the number (multip...

We investigate the KNO scaling function of the modified negative binomial distribution (MNBD), because this MNBD can explain the oscillating behaviors of the cumulant moment observed in e+e− annihilations and in hadronic collisions. By using a straightforward method and the Poisson transform we derive the KNO scaling function from the MNBD. The KNO form of experimental data in e+e− collisions a...

Fractional derivative in time variable is introduced into the Fokker-Planck equation of a population growth model. It’s solution, the KNO scaling function, is transformed into the generating function for the multiplicity distribution. Formulas of the factorial moment and the Hj moment are derived from the generating function, which reduces to that of the negative binomial distribution (NBD), if...

In comparing the mean count of two independent samples, some practitioners would use the t-test or the Wilcoxon rank sum test while others may use methods based on a Poisson model. It is not uncommon to encounter count data that exhibit overdispersion where the Poisson model is no longer appropriate. This paper deals with methods for overdispersed data using the negative binomial distribution r...