Negative Binomial and Multinomial States: probability distributions and coherent states

Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the $su(2)$ coherent states, we propose ``interpretation'' of $su(1,1)$ and $su(r,1)$ coherent states ``in terms of probability theory''. They will be called the ``negative binomial'' (``multinomial'') ``states'' which correspond to the ``negative'' binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory.Comment: 24 pages, latex, no figure

Generalized binomial distribution in photon statistics

The photon-number distribution between two parts of a given volume is found for an arbitrary photon statistics. This problem is related to the interaction of a light beam with a macroscopic device, for example a diaphragm, that separates the photon flux into two parts with known probabilities. To solve this problem, a Generalized Binomial Distribution (GBD) is derived that is applicable to an arbitrary photon statistics satisfying probability convolution equations. It is shown that if photons obey Poisson statistics then the GBD is reduced to the ordinary binomial distribution, whereas in the case of Bose-Einstein statistics the GBD is reduced to the Polya distribution. In this case, the photon spatial distribution depends on the phase-space volume occupied by the photons. This result involves a photon bunching effect, or collective behavior of photons that sharply differs from the behavior of classical particles. It is shown that the photon bunching effect looks similar to the quantum interference effect.Comment: 8 pages, 4 figure

Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels

To each hyperbolic Landau level of the Poincar\'e disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its decomposition as a perturbation of the negative binomial distribution by a finitely-supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we determine the L\'evy-Kintchine decomposition its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-L\'evy measure, we introduce a new infinitely-divisible distribution for which we derive the characteristic function

Correlated Binomial Models and Correlation Structures

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional probabilities. The beta-binomial distribution is derived by a strong symmetric assumption on the conditional correlations. Our derivation clarifies the 'correlation' structure of the beta-binomial distribution. It is also possible to study the correlation structures of other probability distributions of exchangeable (homogeneous) correlated Bernoulli random variables. We study some distribution functions and discuss their behaviors in terms of their correlation structures.Comment: 12 pages, 7 figure

On the forward-backward correlations in a two-stage scenario

It is demonstrated that in a two-stage scenario with elementary Poissonian emitters of particles (colour strings) arbitrarily distributed in their number and average multiplicities, the forward- backward correlations are completely determined by the final distribution of the forward particles. The observed linear form of the correlations then necessarily requires this distribution to have a negative binomial form. For emitters with a negative binomial distribution of the produced particles distributed so as to give the final distribution also of a negative binomial form, the forward-backward correlations have an essentially non-linear form, which disagrees with the experimental data.Comment: 14 pages in LaTex, 1 figure in Postscrip

Limit theorems for sums of independent random variables

Thesis (M.A.)--Boston University.As the introduction to this thesis has described it the significant content of the thesis is a consideration of the more important aspects of the theory of limiting distributions for the distributions associated with sequences of sums of independent random variables. We begin our analysis with the discussion of the relatively common probability law, the binomial probability law. This is defined and related to two further probability laws: the normal law and the Poisson law. It is shown that in the binomial situation when the number, n, of trials approaches infinity and the probability, p, of success at each trial approaches 0 in such a way that the variable lambda = np remains bounded, the Poisson approximation to the binomial is a uniform approximation. The DeMoivre - Laplace Limit theorem enables us to see the relation of the normal law to the binomial law. It states that the binomial distribution converges to the normal distribution in the situation wherein we are holding p constant and allowing n -> infinity. It is also noted that under favorable conditions the Poisson distribution is itself approximated by means of the Normal distribution [TRUNCATED]

Multimodality of the Markov binomial distribution

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal or trimodal. These are useful to analyze the double-peaking results from a PDE reactive transport model from the engineering literature. Moreover, we give a closed form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time $n$.Comment: 15 pages, 3 figure

On the accuracy of the binomial approximation to the distance distribution of codes

The binomial distribution is a well-known approximation to the distance spectra of many classes of codes. We derive a lower estimate for the deviation from the binomial approximatio