The negative contagion reflection of the polya-eggenberger distribution

The Polya-Eggenberger distribution Involves drawing a ball from an urn containing black and white balls and, after each drawing, returning the ball together with s balls of the same color, The model represents positive contagion since the added balls are the same color as the one drawn, See Johnson and Kotz, (1977),

This paper derives and examines the probability distribution which results from the Polya-Eggenberger model with only one change namely, the s additional balls added after each drawing are of the opposite color, producing a negative contagion model.

Formulas in closed form are presented for the probability distribution function, the mean and variance, all binomial moments and, where s is greater than or equal to the number of balls in the urn at start, the mode, A formula for the mode is conjectured where s is less than the number of balls in the urn at start.

Finally, the probability of obtaining k black balls in n drawings is shown in certain instances to be equal to A_nk/n!

where A_nk are the Eulerian numbers.     

Randomization. pro and contra 1

We discuss a stochastic urn model in which there are two urns A and B. B is originally empty and A contains some fixed number of white and black balls. A player selects integers n>O and b>O. Balls are drawn with replacement in A and balls of the same color are put in B as long as the number of white balls in B exceeds (b-1) times the number of black balls in B. Under this condition, the player stops after drawing n+bx balls and is declared to be a winnear if urn B has x black balls. This number of black balls, x, is shown to have the generalized negative binomial distribution     

A q-Pólya urn model and the q-Pólya and inverse q-Pólya distributions

A q-Pólya urn model is introduced by assuming that the probability of drawing a white ball at a drawing varies geometrically, with rate q, both with the number of drawings and the number of white balls drawn in the previous drawings. Then, the probability mass functions and moments of (a) the number of white balls drawn in a specific number of drawings and (b) the number of black balls drawn until a specific number of white balls are drawn are derived. These two distributions turned out to be q-analogs of the Pólya and the inverse Pólya distributions, respectively. Also, the limiting distributions of the q-Pólya and the inverse q-Pólya distributions, as the number of balls in the urn tends to infinity, are shown to be a q-binomial and a negative q-binomial distribution, respectively. In addition, the positive or negative q-hypergeometric distribution is obtained as conditional distribution of a positive or negative q-binomial distribution, given its sum with another positive or negative q-binomial distribution, independent of it.     

Two stochastic models for the geeta distribution

The first stochastic model is based upon two urns A and B, where A contains a fixed number of white and black balls and B is empty. The player selects an integer β ≥, 2 and draws the balls one by one (with replacement) from urn A and balls of the same colour are put in urn B. The process is continued as long as the number of white balls in B exceeds (β-1) times the number of black balls in B. The player stops after drawing β(x-1) balls and is declared to be a winner if urn B has (x-1) black balls. It is shown that x has the Geeta distribution.

Assuming that the mean μ is a function of two parameters θ and β it has been shown that for small changes inthe value of θ there exists a difference-differential equation which leads to the Geeta distribution.     

The estimation of the number of balls put in a sequence of urns

Let X₁,…,X_2n be independent and identically distributed copies of the non-negative integer valued random variable X distributed according to the unknown frequency function f(x). A total of 2n disjoint sequences of urns, each consisting of k urns, are given. X_j balls are placed in urn sequence j (1 ≤ j ≤ 2n). Each ball is placed in an urn of a given sequence with a certain known probability independently of the other balls. The variables X₁,…,X_2n are not observed; rather we observe whether certain pairs of urns are both empty or not. Our object is to estimate the mean μ of the number of balls X. Two different kinds of estimators of μ are investigated. One of the estimators studied is a method of moments type estimator while the other is motivated by the maximum likelihood principle. These estimators are compared on the basis of their asymptotic mean squared error as k tends to infinity. An application of these results to a problem in genetics involved with estimating codon substitution rates is discussed.     

Normal approximation in an urn model with indistinguishable balls

For the Bose-Einstein Statistics, where n indistinguishable balls are distributed in m urns such that all the arrangements are equally likely, define the random variables

M_k = number of urns containing exactly k balls each;

N_k = number of urns containing at least k balls each.

We consider the approximation of the distributions of M_k and N_k by suitable normal distributions, for large but finite m. Estimates are found for the error in the approximation to both the probability mass function and the distribution function in each case. These results apply also to the alternative model where no urn is allowed to be empty. The results are illustrated by some numerical examples.     

A Bayes Theorem Urn Model Optimization Problem

An urn model is a finite collection of indistinguishable urns together with an arbitrary distribution of a finite number of balls (bills) of k colors (denominations) into the urns. A Bayes theorem expectation optimization problem associated with certain urn models is discussed.     

EXCHANGEABLE PAIRS OF BERNOULLI RANDOM VARIABLES,KRAWTCHOUCK POLYNOMIALS,AND EHRENFEST URNS

This paper derives characterizations of bivariate binomial distributions of the Lancaster form with Krawtchouk polynomial eigenfunctions. These have been characterized by Eagleson, and we give two further characterizations with a more probabilistic flavour: the first as sums of correlated Bernoulli variables; and the second as the joint distribution of the number of balls of one colour at consecutive time points in a generalized Ehrenfest urn. We give a self‐contained development of Krawtchouck polynomials and Eagleson’s theorem.     

A randomly reinforced urn

We study an urn containing balls of two or more colors. The urn is sequentially sampled. Each time a ball is extracted from the urn it is reintroduced in it together with a random number of balls of the same color: the distribution of the number of added balls may depend on the color extracted. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. Applications to sequential clinical trials are considered as well as connections with response-adaptive design of experiments in a Bayesian framework.     

A weighted likelihood ratio of two related negative hypergeometric distributions

In this paper we consider some related negative hypergeometric distributions arising from the problem of sampling without replacement from an urn containing balls of different colours and in different proportions but stopping only after some specific number of balls of different colours have been obtained. With the aid of some simple recurrence relations and identities we obtain in the case of two colours the moments for the maximum negative hypergeometric distribution, the minimum negative hypergeometric distribution, the likelihood ratio negative hypergeometric distribution and consequently the likelihood proportional negative hypergeometric distribution. To the extent that the sampling scheme is applicable to modelling data as illustrated with a biological example and, in fact, many situations of estimating Bernoulli parameters for binary traits within a finite population, these are important first-step results.     

Quota fulfillment times

An example of the classical occupancy problem is to sample with replacement from an urn containing several colours of balls and count the number of balls sampled until a given number of “quotas” are filled. This and the corresponding random variable for sampling without replacement will be referred to as quota fulfillment times. Asymptotic and exact methods for computing moments and distributions are given in this paper. Moments of quota fulfillment times are related to moments of order statistics of beta and gamma random variables. Most of the results for sampling without replacement and some of the results for sampling with replacement are believed to be new. Some other known sampling-with-replacement results are given for comparative purposes.     

Generalized Markov-Polya urn models with predetermined strategies

By using combinatorial methods involving lattice path combinatorics, three generalized probability models dependent on predetermined strategies have been obtained with the help of urn models.The models have been developed with the help of a sampling scheme which unifies both, the binomial and the inverse binomial sampling schemes. These models generate a number of important discrete probability distributions both as particular cases and as limiting cases. Recurrence relations among the moments of the models have also been obtained.     

On order statistics from thelog-logistic distribution and their properties

The aim of this paper is to derive the exact forms of the p.d.f. and the moments of the rth order statistics in a sample of size n from the Log-logistic (L_l ) distribution. Measures of skewness and kurtosis are tabulated. The recurrence relations between the moments of all order statistics and an expression of the covariance between any two order statistics, x_i and x_jand the distribution of the ratio of X_i to x_j are derived.     

Generalized polya eggenberger family of distributions and its relation to lagrangian katz family

Janardan (1973) introduced the generalized Polya Eggenberger family of distributions (GPED) as a limiting distribution of the generalized Markov-Polya distribution (GMPD). Janardan and Rao (1982) gave a number of characterizing properties of the generalized Markov-Polya and generalized Polya Eggenberger distributions. Here, the GPED family characterized by four parameters, is formally defined and studied. The probability generating function, its moments, and certain recurrence relations with the moments are provided. The Lagrangian Katz family of distributions (Consul and Famoye (1996)) is shown to be a sub-class of the family of GPED (or GPED ₁ ) as it is called in this paper). A generalized Polya Eggenberger distribution of the second kind (GPED ₂ ) is also introduced and some of it's properties are given. Recurrence relations for the probabilities of GPED ₁ and GPED ₂ are given. A number of other structural and characteristic properties of the GPED ₁ are provided, from which the properties of Lagrangian Katz family follow. The parameters of GMPD ₁ are estimated by the method of moments and the maximum likelihood method. An application is provided.     

Robust Bayesian analysis of the binomial empirical Bayes problem

Empirical Bayes estimation is considered for an i.i.d. sequence of binomial parameters θ_i arising from an unknown prior distribution G(.). This problem typically arises in industrial sampling, where samples from lots are routinely used to estimate the lot fraction defective of each lot. Two related issues are explored. The first concerns the fact that only the first few moments of G are typically estimable from the data. This suggests consideration of the interval of estimates (e.g., posterior means) corresponding to the different possible G with the specified moments. Such intervals can be obtained by application of well-known moment theory. The second development concerns the need to acknowledge the uncertainty in the estimation of the first few moments of G. Our proposal is to determine a credible set for the moments, and then find the range of estimates (e.g., posterior means) corresponding to the different possible G with moments in the credible set.     

Exact Distributions of the Number of r-Records and the r-Record and Inter-r-Record Times

Consider a sequence of independent and identically distributed random variables with an absolutely continuous distribution function. The probability functions of the numbers K_n,r and N_n,r of r-records up to time n of the first and second type, respectively, are obtained in terms of the non central and central signless Stirling numbers of the first kind. Also, the binomial moments of K_n,r and N_n,r are expressed in terms of the non central signless Stirling numbers of the first kind. The probability functions of the times L_k,r and T_k,r of the kth r-record of the first and second type, respectively, are deduced from those of K_n,r and N_n,r. A simple expression for the binomial moments of T_k,r is derived. Finally, the probability functions and binomial moments of the kth inter-r-record times U_k,r = L_k,r ? L_k?1,r and W_k,r = T_k,r ? T_k?1,r are obtained as sums of finite number of terms.     

Statistical control charts for shifted generalized poisson distribution

Summary The use of shifted (or zero-truncated) generalized Poisson distribution to describe the occurrence of events in production processes is considered. The methods of moments and maximum likelihood are proposed for estimating the parameters of shifted generalized Poisson distribution. Control charts for the total number of events and for the average number of events are developed. Finally, a numerical example is used to illustrate the construction of control charts.     

A generalized binomial distribution

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.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models