Related papers: A probabilistic proof of a binomial identity

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…

We examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its Poisson-like limit. Such generalizations might be found in quantum optics with imperfect detection. We…

We present an elementary combinatorial proof of the celebrated Friendship theorem. The proof involves looking at independent sets and constructing a bound on their size which forces a contradiction.

A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

The estimation of a probability p from repeated Bernoulli trials is considered in this paper. A sequential approach is followed, using a simple stopping rule. A closed-form expression and an upper bound are obtained for the mean absolute…

We prove that a random bivariate polynomial with plus minus 1 coefficients is irreducible with high probability.

We present new proofs for some summation identities involving Stirling numbers of both first and second kind. The two main identities show a connection between Stirling numbers and Bessel numbers. Our method is based on solving a particular…

We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of…

A central problem in proof-theory is that of finding criteria for identity of proofs, that is, for when two distinct formal derivations can be taken as denoting the same logical argument. In the literature one finds criteria which are…

We prove some combinatorial identities using the Polya urn and the closely related Hoppe urn.

We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity {\em not} involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a…

We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…

We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial…

In here, I present a series of combinatorial equalities derived using a graph based approach. Different nodes in the graphs are visited following probabilistic dynamics of a moving dot. The results are presented in such a way that the…

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.

In this paper, we establish a fundamental connection between binomial parameters and means of bounded random variables. Such connection finds applications in statistical inference of means of bounded variables.

In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.

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.…

Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely…