Consul
(1974) derived a quasi-binomial distribution (QBD) by
considering a simple urn model with a predetermined
strategy. Consul and Mittal (1975) derived another form
of QBD, using a new predetermined strategy. Janardan
(1975) discussed Markov Polya urn models dependent on
a predetermined strategy. Das (1993) obtained a class
of QBDs with the aid of Abel's generalization of the
binomial formula and a class of generalized poisson
distributions (GPDs) as a limiting form of QBDs. Following
these authors, here we have obtained a class of QBDs
of order k by considering Abel's generalization of the
binomial formula. Further, a class of GPDs of order
k is also defined as a limiting case of QBDs of order
k.
Let
k be a positive integer. Suppose we are given
independent trials with success probability p. The
distribution of the number of occurrences of consecutive
k-successes until the nth trial
is called the binomial distribution of order k
and is denoted by  .
Consul (1974) considered a simple urn model with a predetermined
strategy and derived the Quasi-Binomial Distribution
(QBD). Das (1993) defined a class of QBD by considering
Abel's generalization of the binomial formula (Riordon,
1968, p. 18).
Every
distribution of order 1 is of course the usual corresponding
discrete distribution. This is the only explicit relation
between distributions of order k and that of
order 1.
In
this paper, we find out a class of QBDs of order k
using Abel's generalization of the binomial formula.
Further, a class of Generalized Poisson distributions
(GPDs) of the order k is defined as a limiting
form of a class of QBDs of order k. Different
distributional properties of QBDs and GPDs of order
k are studied. |
