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lecturenotes235A - MATH/STAT 235A Probability Theory...

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Unformatted text preview: MATH/STAT 235A Probability Theory Lecture Notes, Fall 2011 Dan Romik Department of Mathematics, UC Davis November 11, 2011 Contents Chapter 1: Introduction 5 1.1 What is probability theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The algebra of events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 2: Probability spaces 10 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 3: Random variables 16 3.1 Random variables and their distributions . . . . . . . . . . . . . . . . . . . . 16 3.2 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 4: Random vectors and independence 23 4.1 Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Multi-dimensional distribution functions . . . . . . . . . . . . . . . . . . . . 24 4.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 5: The Borel-Cantelli lemmas 29 Chapter 6: A brief excursion into measure theory 31 Chapter 7: Expected values 33 7.1 Construction of the expectation operator . . . . . . . . . . . . . . . . . . . . 33 7.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.3 Integration to the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.4 Computing expected values . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.5 Expectation and independent random variables . . . . . . . . . . . . . . . . 40 7.6 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter 8: Laws of large numbers 44 8.1 Weak laws of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 8.2 Strong laws of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2 Chapter 9: Applications and further examples 52 9.1 The Weierstrass approximation theorem . . . . . . . . . . . . . . . . . . . . 52 9.2 Infinite expectations and triangular arrays . . . . . . . . . . . . . . . . . . . 53 9.3 Random series of independent samples . . . . . . . . . . . . . . . . . . . . . 57 Chapter 10: Stirlings formula and the de Moivre-Laplace theorem 61 Chapter 11: Convergence in distribution 65 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 11.3 Compactness and tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Chapter 12: Characteristic functions 71 12.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 71 12.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 12.3 The inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12....
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