Fundamentals of Mathematical Statistics: Probability for Statistics

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Springer Science & Business Media, Aug 17, 1989 - Mathematics - 432 pages
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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
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About the author (1989)

Nguyen, New Mexico State University, Las Cruces