Showing papers on "Probability mass function published in 1986"

Goodness of fit tests for discrete distributions

Abstract: Some alternative procedures for testing goodness of fit in discrete distributions are discussed here.. These procedures are based on the probability generating functions.. The methods considered are quite general, being applicable in multidimensional situations., The strength of the tests lies in that no ambiguity as to classification of the data arises.. Hov-ever, some difficulties in the proposed procedures are also pointed out.

Apparatus and method for estimating, from sparse data, the probability that a particular one of a set of events is the next event in a string of events

Abstract: Apparatus and method for evaluating the likelihood of an event (such as a word) following a string of known events, based on event sequence counts derived from sparse sample data. Event sequences--or m-grams--include a key and a subsequent event. For each m-gram is stored a discounted probability generated by applying modified Turing's estimate, for example, to a count-based probability. For a key occurring in the sample data there is stored a normalization constant which preferably (a) adjusts the discounted probabilities for multiple counting, if any, and (b) includes a freed probability mass allocated to m-grams which do not occur in the sample data. To determine the likelihood of a selected event following a string of known events, a "backing off" scheme is employed in which successively shorter keys (of known events) followed by the selected event (representing m-grams) are searched until an m-gram is found having a discounted probability stored therefor. The normalization constants of the longer searched keys--for which the corresponding m-grams have no stored discounted probability--are combined together with the found discounted probability to produce the likelihood of the selected event being next.

Construction of a Probability Distribution from a Fuzzy Information

Abstract: Let X be a variable taking its values on a finite set U. If we have a fuzzy information about these values, then we may represent this information by means of a possibility distribution; but there are some cases in which we need a probability distribution. The aim of this paper is to provide methods to construct such probability distribution. The link between these two kinds of informations, possibilistic and probabilistic, is given by the concept of possibility-probability consistency. The maximum entropy principle is used, being obtained a fuzzy mathematical programming problem, that is solved in several examples.

Probability and Analysis

Abstract: S: 2013 SUMMER VIGRE REU All concepts mentioned in the abstracts will be carefully defined APPRENTICE PROGRAM, weeks 1–5 Madhur Tulsiani and Laci Babai Linear algebra and discrete structures The course will develop the usual topics of linear algebra and illustrate them on (often striking) applications to discrete structures. Emphasis will be on creative problem solving and discovery. The basic topics include determinants, linear transformations, the characteristic polynomial, Euclidean spaces, orthogonalization, the Spectral Theorem, Singular Value Decomposition. Application areas to be highlighted include spectral graph theory (expansion, quasirandom graphs, Shannon capacity), random walks, clustering high-dimensional data, extremal set theory, and more. FULL PROGRAM, weeks 1–8 Probability and Analysis Gregory Lawler, weeks 1-2 Random Walk and the Heat Equation Two closely related topics are random walks and heat flow. One sees this by considering heat as consisting of a large (infinite number?) of ”heat particles” all moving randomly and independently. I will present some of the mathematics that makes this rigorous. I start with random walk in the integer lattice and then use this to give a model for discrete heat flow. Analysis of this flow will use tools from linear algebra. In the second week I will consider the continuous analogues: the random walk become Brownian motion; the discrete heat equation because a partial differential equation (called, amazingly, the heat equation!), and the linear algebra argument is replaced with one using Fourier series. This course has been given before and the notes have been published in a book ”Random Walk and the Heat Equation” published in the Student Mathematical Library series by the American Mathematical Society. Antonio Auffinger, weeks 3-4 Card Shuffling, Branching and beyond The goal of this two week course is to present some topics which are accessible to advanced undergraduates yet are areas of research in probability. We will first discuss the problem of shuffling a deck of cards. We will start by studying random

Frequency analysis of low flows: hypothetical distribution methods and a physically based approach

Abstract: A mixed log Pearson type III distribution, a double bounded probability density function, partial duration series and a physically based approach are analyzed for frequency estimates of low flows. The mixed log Pearson III involves a point probability mass at zero for intermittent streams. The double bounded probability distribution has lower and upper bounds with a point mass at the lower bound. Two approaches are used in partial duration series i) truncation, and ii) censoring which represent curtailing of the population and the sample respectively. The parameters are estimated by maximum likelihood procedure. Considering low flows as part of the recession limb of stream flow hydrographs a physically based approach is formulated. By using the exponential decay of stream recessions and considering the initial recession flows, recession durations, and recharge due to incoming storms as statistically independent random variables, a first order random coefficient Markov model for initial recession flows is formed. The resulting steady state probability distribution for initial recession flows is combined with the probability distribution of the exponential decay to obtain the probabilities of low flow events. The methods are applied to both perennial and intermittent streams.

Multiparameter estimation for some multivariate discrete distributions with possibly dependent components

Abstract: In multiparameter estimation for multivariate discrete distributions with infinite support, inadmissibility problems in situations where the multivariate probability distribution function isnot a product of the one-dimensional marginal probability distribution functions have previously been unexplored. This paper examines the inadmissibility problem in some of these situations. Special attention is given to estimating the mean of a negative multinomial distribution. In estimating the mean vector, certain Clevenson-Zidek type estimators are shown to be uniformly better than the usual estimator under a large class of generally scaled squared loss functions. Some of the results are generalized to other multivariate discrete distributions and to situations where several independent negative multinomial distributions are considered.

Characterization of the quantitative-qualitative measure of inaccuracy for discrete generalized probability distributions

Abstract: A characterization of the quantitative-qualitative measure of inaccuracy depending upon the additivity postulate and the mean value property for discrete generalized probability distribution has been provided.

Studies in the history of probability and statistics XLI Waring and Sylvester on random algebraic equations

Abstract: Modern research on random algebraic equations was initiated by Offord & Littlewood (1938), and has been concerned mainly with the probability distribution of the number of real roots of the equation xaixi = 0, where the ai are independent identically distributed real random variables. It is known that if they are normal with mean P * 0, the expectation of the number of real roots is asymptotically i'log n, while if pu = 0 it is 27-1 log n. Much more is known but there is unfortunately no monograph on the subject. Earlier than this, both Waring and Sylvester used probability in their discussion of algebraic equations. Their writings are of interest as early examples of the use of probabilitistic and statistical ideas in looking at pure mathematical problems. They involve studies of the Rule of Signs, stated without proof by Newton in his Lectures on Algebra, published as the Arithmetica Universalis (1707). Both writers present problems of interpretation. The rule was expressed by Sylvester (1864) as follows: we associate with the algebraic equation

Steady-state probability density function of the phase error for a DPLL with an integrate-and-dump device

Abstract: The steady-state behavior of a particular type of digital phase-locked loop (DPLL) with an integrate-and-dump circuit following the phase detector is characterized in terms of the probability density function (pdf) of the phase error in the loop. Although the loop is entirely digital from an implementation standpoint, it operates at two extremely different sampling rates. In particular, the combination of a phase detector and an integrate-and-dump circuit operates at a very high rate whereas the loop update rate is very slow by comparison. Because of this dichotomy, the loop can be analyzed by hybrid analog/digital (s/z domain) techniques. The loop is modeled in such a general fashion that previous analyses of the Real-Time Combiner (RTC), Subcarrier Demodulator Assembly (SDA), and Symbol Synchronization Assembly (SSA) fall out as special cases.

Reflexive pair production probability study for equal spheres in a saturated random distribution in three dimensions

Abstract: A study of reflexive pair formation probability for a three‐dimensional random distribution of spheres. The numerical density (0.704 sphere centers/unit volume) of the distributions studied is approximately the previously known saturation density of 0.709. A probability of 0.517±0.004 was found for the spheres. This can be compared to the known value of 0.5926 for the Poisson distribution points. Apparently the probability of formation of reflexive pairs is inhibited when the entities being studied possess a finite impenetrable volume. Coordination numbers for reflexive and nonreflexive pair members are found to be essentially the same.