Generalized Hypergeometric Factorial Moment Distribution of Order k

Abstract

Here we introduce the generalized hypergeometric functional moment distribution of order k (GHFMD (k)) in the distribution of the random sum ${\sum }_{j=1}^N{X}_j\hspace{0.17em}\hspace{0.17em}\text{of}\hspace{0.17em}{X}_j\text{s},$ having Hira no's k-point distribution, where N, independent of X _j 's, has the generalized hypergeomet ric factorial moment distribution. Well-known discrete distributions of order k such as cluster binomial, cluster negative binomial and extended Poisson are shown to be special cases of GHFMD(k). The probability mass function, recurrence relations for probabilities and factorial moments of GHFMD (k) are found out. The beta or the gamma mixture of GHFMD (k) is shown to be a GHFMD (k). Finally GHFMD (k) is obtained as a limit of another GHFMD (k).

AMS (2000) Subject Classification: Primary 60E05, 60E10; Secondary 33C20.