Probabilities and moments of generalized discrete distributfons using finite difference operators

The probabilities and factorial moments of the univar iate and multivariate generalized (or compound) discrete di st r-Lbut Lons with probability generating functions H(t)=F(G(t)) and H(t₁,…,t_k)=F(G(t₁,…,t_k))or H(t₁,…,t_k) = F(G₁(t₁),…, G_k( t_k)) are derived using finite difference operators.     

Abel series distributions with applications to fluctuations of sample functions of stochastic processes

A two-parameter class of discrete distributions, Abel series distributions, generated by expanding a suitable pa,rametric function into a series of Abel polynomials is discussed. An Abel series distribution occurs in fluctuations of sample functions of stochastic processes and has applications in insurance risk, queueing, dam and storage processes. The probability generating function and the factorial moments of the Abel series distributions are obtained in closed forms. It is pointed out that the name of the generalized Poisson distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that this generalized Poisson distribution is the only member of the Abel series distributions which is closed under convolution.     

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.     

The q-Bernstein basis as a q-binomial distribution

The q-Bernstein basis, used in the definition of the q-Bernstein polynomials, is shown to be the probability mass function of a q-binomial distribution. This distribution is defined on a sequence of zero–one Bernoulli trials with probability of failure at any trial increasing geometrically with the number of previous failures. A modification of this model, with the probability of failure at any trial decreasing geometrically with the number of previous failures, leads to a second q-binomial distribution that is also connected to the q-Bernstein polynomials. The q-factorial moments as well as the usual factorial moments of these distributions are derived. Further, the q-Bernstein polynomial B_n(f(t),q;x) is expressed as the expected value of the function f([X_n]_q/[n]_q) of the random variable X_n obeying the q-binomial distribution. Also, using the expression of the q-moments of X_n, an explicit expression of the q-Bernstein polynomial B_n(f_r(t),q;x), for f_r(t) a polynomial, is obtained.     

Designing communication support systems for strategic planning

While as a distinct and intermittent managerial activity planning is slowly dying, in a systematic and continuous context it is rapidly growing! More and more organizations are beginning to realize that planning entails ongoing learning and adaptation of people rather than one-shot, pseudo-scientific analysis of abstract problems. To facilitate this fundamental transformation in organizational thinking, new electronic management support systems are being created. These systems will facilitate collaborative problem exploration through improved managerial communication. In effect, they will actively assist managers to understand and manage the relationships between the strategic planning process and the other corporate processes such as budgeting, capital investment, performance evaluation and employee compensation.     

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.     

Discrete q-distributions on Bernoulli trials with a geometrically varying success probability

Consider a sequence of independent Bernoulli trials and assume that the odds of success (or failure) or the probability of success (or failure) at the ith trial varies (increases or decreases) geometrically with rate (proportion) q, for increasing i=1,2,…. Introducing the notion of a geometric sequence of trials as a sequence of Bernoulli trials, with constant probability, that is terminated with the occurrence of the first success, a useful stochastic model is constructed. Specifically, consider a sequence of independent geometric sequences of trials and assume that the probability of success at the jth geometric sequence varies (increases or decreases) geometrically with rate (proportion) q, for increasing j=1,2,…. On both models, let X_n be the number of successes up the nth trial and T_k (or W_k) be the number of trials (or failures) until the occurrence of the kth success. The distributions of these random variables turned out to be q-analogues of the binomial and Pascal (or negative binomial) distributions. The distributions of X_n, for n→∞$n\to \infty$, and the distributions of W_k, for k→∞$k\to \infty$, can be approximated by a q  -Poisson distribution. Also, as k→0$k\to 0$, a zero truncated negative q  -binomial distribution _Uk=_Wk|_Wk>0$U_k=W_k|W_k>0$ can be approximated by a q-logarithmic distribution. These discrete q-distributions and their applications are reviewed, with critical comments and additions. Finally, consider a sequence of independent Bernoulli trials and assume that the probability of success (or failure) is a product of two sequences of probabilities with one of these sequences depending only the number of trials and the other depending only on the number of successes (or failures). The q-distributions of the number X_n of successes up to the nth trial and the number T_k of trials until the occurrence of the kth success are similarly reviewed.     

Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number _Nn$N_n$ of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q   being the inverse of the proportion λ$\lambda$ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables _Nn$N_n$, n=1,2,…,$n=1,2,\dots ,$ are deduced. As a corollary the probability function of the time _Tk$T_k$ of the kth record is also expressed in terms of the signless q  -Stirling numbers of the first kind. The mean of _Tk$T_k$ is obtained as a q  -series with terms of alternating sign. Finally, the probability function of the inter-record time _Wk=_Tk-_Tk-1$W_k=T_k-T_{k-1}$ is obtained as a sum of a finite number of terms of q  -numbers. The mean of _Wk$W_k$ is expressed by a q-series. As k   increases to infinity the distribution of _Wk$W_k$ converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.     

Distributions and moments of record values in a sequence of maximally dependent random variables

A sequence of possibly dependent random variables is maximally dependent if all the sample maxima in the sequence have stochastically maximal distributions in the class of all distributions with the same marginals. For a sequence of maximally dependent standard uniform random variables, we determine the distribution functions of record times and values. We show that the distribution of the record occurrence times coincides with the respective distribution for the i.i.d. sequence, and the distributions of the record values are stochastically maximal in the class of sequences with the same record times distributions, containing all the exchangeable sequences. We also derive analytic formulae for the moments of record values from the maximally dependent sequence, and compare them with those of the i.i.d. case.