Basic Concepts of Probability and Statistics

Description

Second Edition. — Society for Industrial and Applied Mathematics, Philadelphia, PA- 2005. — 478 p.Basic Concepts fills the need for an introduction to the fundamental ideas of modem statistics that was mathematically rigorous but did not require calculus. This was achieved by restricting attention to discrete situations. The book was translated into Italian, Hebrew, Danish, and, more recently, Farsi. The book in many ways is modem in outlook. This is particularly true for its emphasis on models and model-building but also by its coverage of such topics as survey sampling (both simple and stratified), experimental design (with a proof of the superiority of factorial design over varying one factor at a time), its presentation of nonparametric tests such as the Wilcoxon, and its discussion of power (including the Neyman—Pearson lemma). The book is very much in the spirit of texts on discrete mathematics and could well be used to supplement high school and college courses on this subject.Although the book contains a large number of examples from a great variety of fields of application, it does not base these on real data. When used as a textbook, this drawback could be remedied by adding a laboratory in which actual situations are discussed.Probability models.Random experimentsEmpirical basis of probabilitySimple events and their probabilitiesDefinition of a probability modelUniform probability modelsThe algebra of eventsSome laws of probabilitySampling.A model for samplingThe number of samplesOrdered samplingSome formulas for samplingProduct models.Product models for two-part experimentsRealism of product modelsBinomial trialsThe use of random numbers to draw samplesConditional probability.The concept of conditional probabilityIndependenceTwo-stage experimentsProperties of two-stage models; Bayes' lawApplications of probability to geneticsMarriage of relativesPersonal probabilityRandom variables.DefinitionsDistribution of a random variableExpectationProperties of expectationLaws of expectationVarianceLaws of varianceSpecial distributions.The binomial distributionThe hypergeometric distributionStandard unitsThe normal curve and the central limit theoremThe normal approximationThe Poisson approximation for np = 1The Poisson approximation: general caseThe uniform and matching distributionsThe law of large numbersSequential stoppingMultivariate distributions.Joint distributionsCovariance and correlationThe multinomial distributionEstimation.Unbiased estimationThe accuracy of an unbiased estimateSample sizeEstimation in measurement and sampling models.A model for samplingA model for measurementsComparing two quantities or populationsEstimating the effect of a treatmentEstimation of varianceOptimum methods of estimation.Choice of an estimateExperimental designStratified samplingAn inequality and its applicationsTests of significance.First concepts of testingMethods for computing significanceThe chi-square test for goodness of fitTests for comparative experiments.Comparative experimentsThe Fisher-Irwin test for two-by-two tablesThe Wilcoxon two-sample testThe Wilcoxon distributionThe sign test for paired comparisonsWilcoxon‘s test for paired comparisonsThe problem of tiesThe concept of power.The two kinds of errorDetermination of sample size. Power of a testThe power curveOne- and two-sided testsChoice of test statisticThe X, t and Wilcoxon one-sample testsTables.Number of combinationsBinomial distributionsSquare rootsNormal approximationPoisson approximationChi-square approximationWilcoxon two-sample distributionWilcoxon paired-comparison distribution.