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n1 - Stat 5101 Lecture Notes Charles J Geyer Copyright 1998...

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Unformatted text preview: Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000 by Charles J. Geyer January 16, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.3 Random Variables: Informal Intuition . . . . . . . . . . . 3 1.1.4 Random Variables: Formal Definition . . . . . . . . . . . 3 1.1.5 Functions of Random Variables . . . . . . . . . . . . . . . 7 1.2 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 General Definition . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Discrete Random Variables . . . . . . . . . . . . . . . . . 9 1.2.3 Continuous Random Variables . . . . . . . . . . . . . . . 12 1.3 Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Discrete Random Vectors . . . . . . . . . . . . . . . . . . 15 1.3.2 Continuous Random Vectors . . . . . . . . . . . . . . . . 15 1.4 The Support of a Random Variable . . . . . . . . . . . . . . . . . 17 1.5 Joint and Marginal Distributions . . . . . . . . . . . . . . . . . . 18 1.6 Multivariable Change of Variables . . . . . . . . . . . . . . . . . 22 1.6.1 The General and Discrete Cases . . . . . . . . . . . . . . 22 1.6.2 Continuous Random Vectors . . . . . . . . . . . . . . . . 22 2 Expectation 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . 32 2.3 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Axioms for Expectation (Part I) . . . . . . . . . . . . . . 32 2.3.2 Derived Basic Properties . . . . . . . . . . . . . . . . . . . 34 2.3.3 Important Non-Properties . . . . . . . . . . . . . . . . . . 36 2.4 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 First Moments and Means . . . . . . . . . . . . . . . . . . 38 2.4.2 Second Moments and Variances . . . . . . . . . . . . . . . 40 2.4.3 Standard Deviations and Standardization . . . . . . . . . 42 2.4.4 Mixed Moments and Covariances . . . . . . . . . . . . . . 43 2.4.5 Exchangeable Random Variables . . . . . . . . . . . . . . 50 2.4.6 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 50 iii iv Stat 5101 (Geyer) Course Notes 2.5 Probability Theory as Linear Algebra . . . . . . . . . . . . . . . 55 2.5.1 The Vector Space L 1 . . . . . . . . . . . . . . . . . . . . . 56 2.5.2 Two Notions of Linear Functions . . . . . . . . . . . . . . 58 2.5.3 Expectation on Finite Sample Spaces . . . . . . . . . . . 59 2.5.4 Axioms for Expectation (Part II) . . . . . . . . . . . . . . 62 2.5.5 General Discrete Probability Models . . . . . . . . . . . . 64 2.5.6 Continuous Probability Models . . . . . . . . . . . . . . . 66 2.5.7 The Trick of Recognizing a Probability Density . . . . . . ....
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n1 - Stat 5101 Lecture Notes Charles J Geyer Copyright 1998...

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