Related papers: On Probabilistic Proofs of Certain Binomial Identi…
While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…
In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof…
A random phenomenon may have two sources of random variation: an unstable identity and a set of external variation-generating factors. When only a single source is active, two mutually exclusive extreme scenarios may ensue that result in…
In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical…
The proof identity problem asks: When are two proofs the same? The question naturally occurs when one reflects on mathematical practice. The problem understandably can be seen as a challenge for mathematical logic, and indeed various…
We describe a novel way to represent the probability distribution of a random binary string as a mixture having a maximally weighted component associated with independent (though not necessarily identically distributed) Bernoulli…
We use a probabilistic approach to describe the behavior as $n -> \infty$ of the Laplace transforms of $P^n$, where $P$ a fixed complex polynomial. As a consequence we obtain a new elementary proof of an result of Gillis-Ismail-Offer in the…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here…
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
The paper presents some distributional properties of logistic order statistics subject to independent exponential one-sided and two-sided shifts. Utilizing these properties, we extend several known results and obtain some new…
Testing whether the observed data conforms to a purported model (probability distribution) is a basic and fundamental statistical task, and one that is by now well understood. However, the standard formulation, identity testing, fails to…
The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…
A logic is defined that allows to express information about statistical probabilities and about degrees of belief in specific propositions. By interpreting the two types of probabilities in one common probability space, the semantics given…
It is known that Bernoulli scheme of independent trials with two outcomes is connected with the binomial coefficients. The aim of this paper is to indicate stochastic processes which are connected with the $q$-polynomial coefficients (in…
From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for…
While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. In this paper we study the question of uniqueness of the result (unique limit distribution), and develop a…
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…
The idea of fully accepting statements when the evidence has rendered them probable enough faces a number of difficulties. We leave the interpretation of probability largely open, but attempt to suggest a contextual approach to full belief.…