# Fundamentals of Mathematical Statistics: Probability for Statistics

Springer Science & Business Media, Jul 25, 1989 - Mathematics - 432 pages
This is the first half of a text for a two semester course in mathematical statistics at the senior/graduate level for those who need a strong background in statistics as an essential tool in their career. To study this text, the reader needs a thorough familiarity with calculus including such things as Jacobians and series but somewhat less intense familiarity with matrices including quadratic forms and eigenvalues. For convenience, these lecture notes were divided into two parts: Volume I, Probability for Statistics, for the first semester, and Volume II, Statistical Inference, for the second. We suggest that the following distinguish this text from other introductions to mathematical statistics. 1. The most obvious thing is the layout. We have designed each lesson for the (U.S.) 50 minute class; those who study independently probably need the traditional three hours for each lesson. Since we have more than (the U.S. again) 90 lessons, some choices have to be made. In the table of contents, we have used a * to designate those lessons which are "interesting but not essential" (INE) and may be omitted from a general course; some exercises and proofs in other lessons are also "INE". We have made lessons of some material which other writers might stuff into appendices. Incorporating this freedom of choice has led to some redundancy, mostly in definitions, which may be beneficial.

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### Contents

 ELEMENTARY PROBABILITY AND STATISTICS 1 LESSON 1 RELATIVE FREQUENCY 3 LESSON 2 SAMPLE SPACES 11 LESSON 3 SOME RULES ABOUT SETS 19 LESSON 4 THE COUNTING FUNCTION FOR FINITE SETS 27 LESSON 5 PROBABILITY ON FINITE SAMPLE SPACES 36 LESSON 6 ORDERED SELECTIONS 43 LESSON 7 UNORDERED SELECTIONS 49
 LESSON 6 SOME ALGEBRA OF RANDOM VARIABLES 226 LESSON 7 CONVERGENCE OF SEQUENCES OF RANDOM VARIABLES 233 LESSON 8 CONVERGENCE ALMOST SURELY AND IN PROBABILITY 241 LESSON 9 INTEGRATION I 249 LESSON 10 INTEGRATION II 256 LESSON 11 THEOREMS FOR EXPECTATION 266 LESSON 12 STIELTJES INTEGRALS 275 LESSON 13 PRODUCT MEASURES AND INTEGRALS 285

 LESSON 8 SOME UNIFORM PROBABILITY SPACES 57 LESSON 9 CONDITIONAL PROBABILITYINDEPENDENCE 63 LESSON 10 BAYES RULE 72 LESSON 11 RANDOM VARIABLES 77 LESSON 12 EXPECTATION 84 LESSON 13 A HYPERGEOMETRIC DISTRIBUTION 93 LESSON 14 SAMPLING AND SIMULATION 99 LESSON 15 TESTING SIMPLE HYPOTHESES 104 LESSON 16 AN ACCEPTANCE SAMPLING PLAN 115 LESSON 17 THE BINOMIAL DISTRIBUTION 123 LESSON 18 MATCHING AND CATCHING 140 LESSON 19 CONFIDENCE INTERVALS FOR A BERNOULLI 0 150 LESSON 20 THE POISSON DISTRIBUTION 166 LESSON 21 THE NEGATIVE BINOMIAL DISTRIBUTION 177 PROBABILITY AND EXPECTATION 184 LESSON 1 SOME SET THEORY 186 LESSON 2 BASIC PROBABILITY THEORY 194 LESSON 3 THE CUMULATIVE DISTRIBUTION FUNCTION 201 LESSON 4 SOME CONTINUOUS CDFs 208 LESSON 5 THE NORMAL DISTRIBUTION 216
 LIMITING DISTRIBUTIONS 292 DISCRETE 294 DISCRETE 303 CONTINUOUS 313 CONTINUOUS 322 Lesson 5 Expectation Examples 334 LESSON 6 CONVERGENCE IN MEAN EM DISTRIBUTION 340 LESSON 7 OTHER RELATIONS IN MODES OF CONVERGENCE 349 LESSON 8 LAWS OF LARGE NUMBERS 358 LESSON 9 CONVERGENCE OF SEQUENCES OF DISTRIBUTION FUNCTIONS 366 LESSON 10 CONVERGENCE OF SEQUENCES OF INTEGRALS 373 LESSON 11 ON THE SUM OF RANDOM VARIABLES 380 LESSON 12 CHARACTERISTIC FUNCTIONS I 388 LESSON 13 CHARACTERISTIC FUNCTIONS II 398 LESSON 14 CONVERGENCE CHARACTERISTIC FUNCTIONS OF SEQUENCES OF CHARACTERISTIC FUNCTIONS 406 LESSON 15 CENTRAL LIMIT THEOREMS 415 REFERENCES 425 INDEX 427 Copyright