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Random Oper, and Stock. Equ., Vol. 5, No. 3, pp. 227-236 (1997) © VSP 1997 The free creation and annihilation operators as the central limit of the quantum Bernoulli process M. De. GIOSA and Y. G. LU Department of Mathematics, University of Bari, Italy Received for ROSE December 20, 1996 Abstract—We give a new construction of the free creation, annihilation operators by taking the central limit of the quantum Bernoulli process. 1. INTRODUCTION Free central limit theorem was studied originally in the catalogue of operator algebra (see [9,10] and the references

Statistics & Decisions 9,213 - 234 (1991) © R. Oldenbourg Verlag, München 1991 - 0721-2631/91 $3.00+0.00 EMPIRICAL BAYES SELECTION FOR THE HIGHEST SUCCESS PROBABILITY IN BERNOULLI PROCESSES WITH NEGATIVE BINOMIAL SAMPLING TaChen Liang Received: Revised version: January 17, 1991 Aba tract. We study the problem of selecting the highest success probability from among several Bernoulli processes with negative binomial sampling via the nonparametric empirical Bayes approach. A monotone selection rule is proposed on basis of monotone empirical Bayes

://billiards.mathieu.bouville.name/biMar/. KEYWORDS: billiards, Markov process, Bernoulli process, sport, game Author Notes: I wish to thank A. Allenic for his useful comments. I also wish to thank J.-L. Frantz, J.-M. Fray, and R. Jewett who provided me with score sheets. 1. Introduction The word ‘billiards’ can refer to cue games in general —including pool and snooker— or it can refer to a specific game, also called ‘carom billiards’ or ‘carom.’ The latter, to which we refer here, is played with three balls on a pocketless table; it is popular in continental Eu- rope, the Middle-East, and South America. (A short

) (PEtG), a new polyacetal, was synthesized via anionic polymerization in CH2Cl2. End-capping was achieved using phenyl iso- cyanate. Average molecular weights in the range of 9000 - 216 000 were obtained. 1H and 13C NMR characterization highlight the non-stereoregularity of PEtG. The polymer is quasi-atactic and the polymerization almost followed a Bernoulli process. Compared to its homologue poly(methyl glyoxylate), PEtG exhibits enhanced thermal stability (>200°C) and a lower glass transition temperature, Tg. Molecular weight influences Tg, but does not affect

Bayes selection for the highest success probability in Bernoulli processes with negative binomial sampling 213 H. L. Koul, P. K. Sen Weak convergence of a weighted residual empirical process in autoregression 235 S. S. Gupta, TaChen Liang On the asymptotic optimality of certain empirical Bayes simultaneous testing procedures for Poisson populations 263 A. L. Rukhin Admissible estimators of discriminant coefficients 285 Number 4 W. Wefelmeyer Efficient estimation in multiplicative counting process models 301 J. Eichenauer-Herrmann Gamma-minimax estimation

estimator 81 Shi, Z., Empirical process of U-statistic structure 181 Sriram, Τ. N., Second order approximation to the risk of a sequential procedure measured under squared relative error loss 375 TaChen Liang, Empirical Bayes selection for the highest success probability in Bernoulli processes with negative binomial sampling 213 - s. Gupta, S. S„ 263 Wefelmeyer, W., Efficient estimation in multiplicative counting process models 301 Wells, Μ. T., e-Solutions of minimization problems in statistics 177 Book Reviews Kalashnikov, V.V., Penkov, B., Zolotarev, V

distribution is also easy to calculate. By Sec. 5.7 we derive for all i. Since the trials of a Bernoulli process are independent, Sec. 5.7 (5) In the example of Fig. 6.1, the variances are .45, 1.05, 1.25, 1.05, .45, while the standard deviations are .67, 1.02, 1.12, 1.02, .67. We see again that for fixed n, the dispersion of a binomial distribution is smaller as ρ is farther from .5 in either direction. The facts about the distribution of r, the total number of S's in η Bernoulli trials, can be readily extended to the distribution of r/n, the fraction of S's in η

Subject Index A Acts, 16, 52-53 comparison of, under uncertainty, 52-54 Additivity, finite, 57, 95 countable, 95 Algebra, 167, 175 information, 174ff. Β Bayes' decision rules, 22 Bayes theorem, 16-18, 28 applied to Bernoulli process, 145-146 Bayesian procedures, 20ff., 223, 226-227 Bernoulli process, 116ff. parameter of, 116 Bernoulli trials, 113ff. Binomial distribution, 113-116 mean of, 115 variance of, 115 Break-even point, 182 If. C Central limit theorem, 129-131 Certainty equivalent, 184 Chebychev's Inequality, 132-134 Coherence, 44

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of Gambles 90 Appendix B. Satisficing Behavior Rules 91 5. Analysis of Probability 95 5.1 Probability Measures and Random Variables 95 5.2 Sample Space 96 5.3 Probability Distributions 97 5.4 Joint, Conditional and Marginal Probability 98 8 5.5 Statistical Independence 99 5.6 Mathematical Expectation: The Mean 100 5.7 Measures of Dispersion: Variance and Standard Deviation . . . . 104 5.8 Problems 106 Part II. Simple Random Processes and Probability Distributions 6. Bernoulli Trials 113 6.1 Binomial Distributions 113 6.2 Bernoulli Process 116 6

Jalan Ayer Keroh Lama, Melaka, 75450 Malaysia Email: hakim.mellah@mmu.edu.my 1 Alcatel Networks (Malaysia) Multimedia University Faculty of Engineering, Jalan Multimedia 63100 Cyberjaya Selangor Malaysia Received 11 April 2007; accepted 2 July 2007 Fig. 1: The proposed switch model 156 Journal of Optical Communications 28 (2007) 2 identically distributed (iid) Bernoulli process with pa- rameter ρ (0 < ρ ≤ 1) [6]. Further, incoming packets choose to go to any of the N output fibers with equal probability, independent of all other packets (i.e., it chooses a