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An introduction to probability theory

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An introduction to probability theory
Christel Geiss and Stefan Geiss
Department of Mathematics and Statistics
University of Jyv¨skyl¨
a
a
June 3, 2009

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Contents
1 Probability spaces
1.1 Definition of σ-algebras . . . . . . . . . . . . . . . . . . . . .
1.2 Probability measures . . . . . . . . . . . . . . . . . . . . . .
1.3 Examples of distributions . . . . . . . . . . . . . . . . . . .
1.3.1 Binomial distribution with parameter 0 < p < 1 . . .
1.3.2 Poisson distribution with parameter λ > 0 . . . . . .
1.3.3 Geometric distribution with parameter 0 < p < 1 . .
1.3.4 Lebesgue measure and uniform distribution . . . . .
1.3.5 Gaussian distribution on R with mean m ∈ R and
variance σ 2 > 0 . . . . . . . . . . . . . . . . . . . . .
1.3.6 Exponential distribution on R with parameter λ > 0
1.3.7 Poisson’s Theorem . . . . . . . . . . . . . . . . . . .
1.4 A set which is not a Borel set . . . . . . . . . . . . . . . . .

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2 Random variables
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2.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Measurable maps . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Integration
3.1 Definition of the expected value . . . . .
3.2 Basic properties of the expected value . .
3.3 Connections to the Riemann-integral . .
3.4 Change of variables in the expected value
3.5 Fubini’s Theorem . . . . . . . . . . . . .
3.6 Some inequalities . . . . . . . . . . . . .
3.7 Theorem of Radon-Nikodym . . . . . . .
3.8 Modes of convergence . . . . . . . . . . .

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4 Exercises
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4.1 Probability spaces . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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4

CONTENTS

Introduction
Probability theory can be understood as a mathematical model for the intuitive notion of uncertainty. Without probability theory all the stochastic
models in Physics, Biology, and Economics would either not have been developed or would not be rigorous. Also, probability is used in many branches of
pure mathematics, even in branches one does not expect this, like in convex
g...
An introduction to probability theory
Christel Geiss and Stefan Geiss
Department of Mathematics and Statistics
University of Jyv¨askyl¨a
June 3, 2009
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