On the mutual characterization of discrete and continuous 'waiting times'
by
Carlos A. Coelho
Dep. of Mathematics, Institute of Agriculture Technology, Lisbon, Portugal

Let X be the number of failures till the r-th success, being p the probability of success in the population. It is shown that then, if Y given X=x has the distribution of the power 1/b of the 'waiting time' for the (r₁+x)-th event (r₁ > 0) from a Poisson distribution with rate k then the marginal distribution of Y is such that it may be seen as a further extension of the generalized Gamma distribution, which for r₁ > r is the p.d.f of the power 1/b of the sum of two independent random variables, Z₁ and Z₂, where Z₁ has the distribution of the 'waiting time' for the (r₁-r)-th event from a Poisson distribution with rate k and Z₂ has the distribution of the 'waiting time' for the r-th event from a Poisson distribution with rate kp. Moreover, even for b=1, X given Y=y has a distribution which is the generalization of the Hyper-Poisson distribution from Bardwell and Crow (1986) and as such, also a generalization of the Poisson distribution. We will call this distribution the generalized Poisson distribution. But then, the reverse way, when we take this conditional distribution for X, given Y=y, and for Y the marginal distribution above, X will have a Negative Binomial marginal distribution. This result yields a generalization of the result of Greenwood and Yule (1920) on what is sometimes called as the Gamma mixture of Poissons. Examples of application in transportation, epidemiology and agriculture are used for illustration.

Date received: July 22, 2003