A generalized binomial distribution

Abstract

A new generalization of the binomial distribution is introduced that allows dependence between trials, nonconstant probabilities of success from trial to trial, and which contains the usual binomial distribution as a special case. Along with the number of trials and an initial probability of 'success', an additional parameter that controls the degree of correlation between trials is introduced. The resulting class of distributions includes the binomial, unirnodal distributions, and bimodal distributions. Formulas for the moments, mean, and variance of this distribution are given along with a method for fitting the distribution to sample data.

Football fever: goal distributions and non-Gaussian statistics

“…A, where for the additive case of model "A", it is also demonstrated that the continuum limit of P N (n), i.e., N → ∞ with p 0 N and κN kept fixed, is given by the negative binomial distribution (2.4) with r = p 0 /κ and p = 1 − e −κN (note that this also includes the "generalized binomial distribution" considered in Refs. [25,26]). Thus the good fit of a negative binomial distribution to the data can be understood from the "microscopic" effect of self-affirmation of the teams or players, without making reference to the somewhat poorly motivated composition of the pure Poissonian model with a gamma distribution.…”

High-dimensional generation of Bernoulli random vectors

“…Farrell and Sutradhar (2006) use a logistic link function to model the conditional probability of Y i . However, the resulting model is more complex to analyze and suitable for small d. Drezner and Farnum (1993) introduced a Bernoulli process {Y j ; j ≥ 1} in which the random variables Y j are correlated as the success probability of a trail conditional on the previous trials depends on the total number of successes so far. That is,…”

A correlated poisson distribution for correlated events

“…P(r,j) for a given p is the probability of r successes in j trials. The recursive relationship that defines the GBD is (Drezner and Farnum, 1993): along with the starting values P(0,1)= 1-p, P(1,l) =p and boundary conditions P(-lj) =Po'+ l j ) = 0.…”