ABSTRACT
Like the standard beta distribution, (XVII.l) enjoys a simple reflec tive property: if a random variable X has the pdf (XVII.l) then
A random variable X is said to have the compound confluent hypergeometric distribution with parameters a > 0 , 6 > 0, -o o <
and if its pdf is:
exp
where is the confluent hypergeometric function of two variables defined by
It is straightforward to check that the moment generating function of (XVII.l) is:
Since H(a,b,r,s,u,9) is a finite positive real number for all admis sible values of the parameters (Gordy, 2000, Theorem 1), all the moments above must exist. The first moment changes monotonically with s; actually, it can be shown that
The Lagrangian beta distribution arises as the inter-arrival dis tribution for the generalized negative binomial process (Jain and Consul, 1971). Its pdf is:
In the particular case (XVIII. 1) reduces to:
The first and second moments of (XVIII. 1) are:
and
