## Fundamentals of Mathematical Statistics: Probability for StatisticsThis 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 |

425 | |

427 | |

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### Common terms and phrases

a-field approximation Bernoulli Bernoulli trials CDF F consider contains Corollary corresponding countable cumulative distribution function defined Definition density dF(x discrete disjoint distribution function elementary events equal Example Exercise Find finite number following theorem Fubini's Theorem Fx(x given Hence Hint HYPERA hypergeometric distribution hypothesis implies independent indicator function induction INE-Exercises inequality integer interval large numbers least Lebesgue Lebesgue measure lemma Lesson 11 Lesson 9 Let F Let Q liminf limit limsup mathematical mean measurable function non-decreasing non-negative simple functions Note outcomes P(Xj P(Xn pairs pairwise independent parameters Partial proof partition patients points Poisson positive integer probability space properties random variable real numbers real RV real valued function reject H relative frequencies repeated trials right-hand continuous sample space selected Show Similarly statistics subsets Suppose symbol toss uniform distribution uniformly variance Venn diagram X(co