Charalambos A. Charalambides

Bounds on expectation of order statistics from a finite population

Abstract Consider a simple random sample X1,X2,…,Xn, taken without replacement from a finite ordered population Π={x1⩽x2⩽⋯⩽xN} (n⩽N), where each element of Π has equal probability to be chosen in the sample. Let X1:n⩽X2:n⩽⋯⩽Xn:n be the ordered sample. In the present paper, the best possible bounds for the expectations of the order statistics X i : n (1⩽i⩽n) and the sample range Rn=Xn:n−X1:n are derived in terms of the population mean and variance. Some results are also given for the covariance in the simplest case where n=2. An interesting feature of the bounds derived here is that they reduce to some well-known classical results (for the i.i.d. case) as N→∞. Thus, the bounds established in this paper provide an insight into Hartley–David–Gumbel, Samuelson–Scott, Arnold–Groeneveld and some other bounds.

Non-central generalized q-factorial coefficients and q-Stirling numbers

The qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r, at t=0, are thoroughly examined. These numbers for s?0 and s?∞ converge to the non-central q-Stirling numbers of the first and the second kind, respectively. Explicit expressions, recurrence relations, generating functions and other properties of these q-numbers are derived. Further, a sequence of Bernoulli trials is considered in which the conditional probability of success at the nth trial, given that k successes occur before that trial, varies geometrically with n and k. Then, the probability functions of the number of successes in n trials and the number of trials until the occurrence of the kth success are deduced in terms of the qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r.

An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients

An asymptotic formula for the exponential polynomials, Am(x) defined by Σm=0∞Am(x)zm|=exp{x[g(z)−g(0)]} for some (formal) power series g(z), is derived under certain conditions on g(z). Using this formula, the asymptotic normality of the combinatorial distribution p(n; m, λ) = A(m, n)λnAm(λ), n = 0, 1, 2, …, m, 0 < a <λ <b, Am (λ) = ∑n=0m A(m, n)λn , is established. As a special case, the asymptotic normality of a class of combinatorial distributions, where the coefficients A(m, n) satisfy a certain triangular recurrence relation, is concluded.

Derivation of probabilities and moments os ceria1n generalized discrete distributions via urn models

Considering a supply of balls randomly distributed in n distinguishable urns and assuming that qr, r=;0,l,2,…and u r(K) k=1,2 …are the probability function and the factorial moments of the number of balls allocated in any specific urn, the probability function and the factorial moments of certain occupancy distributions are expressed as partition polynomials of qr r=0,l,2,…and u(k), k=l,2,… respectively. In addition the probability function and the factorial moments of these occupancy distributions are given in terms of finite differences of the u-fold convolutions of qr,r=0,l,2,… and u(k), k=1,2,…,respectively. Illustrating these results the probability function and the factorial moments of the n-fold convolution of a zero-truncated discrete distribution and the number of positive random variables given their sum are concluded.. Further This work was completed while the author was visiting lemple University, on leave from the University of Athens

On the Distributions of Absorbed Particles in Crossing a Field Containing Absorption Points

Consider a queue of particles that are required to cross a field containing a random number of absorption points (traps) acting independently. Suppose that if a particle clashes with (contacts) any of the absorption points, it is absorbed (trapped) with probability p and non absorbed with probability q = 1 − p. Let Xn be the number of absorbed particles from a queue of n particles and Tk the number of particles required to cross the field until the absorption of k particles. Assuming that the number Y of absorption points in the field obeys a q-Poisson distribution (Heine or Euler distribution), the distributions of Xn and Tk are obtained as q-binomial and q-Pascal distributions, respectively. Inversely, assuming that Xn obeys a q-binomial distribution (or, equivalently, assuming that Tk obeys a q-Pascal distribution), the distribution of Y is obtained as a q-Poisson distribution (Heine or Euler distribution).

Combinatorial probability interpretation of certain modified orthogonal polynomials

A probabilistic interpretation of a modified Gegenbauer polynomial is supplied by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinatorial interpretations of their coefficients are deduced. Further, probabilistic interpretations of a modified Hermite and a modified Chebyshev polynomial of the first kind are supplied by their expressions in terms of combinatorial probability functions defined on two limiting forms of the compound urn model. Finally, combinatorial interpretations of their coefficients are obtained.