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An Introduction to Probability Theory - Geiss

# An Introduction to Probability Theory - Geiss - An...

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Unformatted text preview: An introduction to probability theory Christel Geiss and Stefan Geiss February 19, 2004 2 Contents 1 Probability spaces 7 1.1 Definition of σ-algebras . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Probability measures . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Examples of distributions . . . . . . . . . . . . . . . . . . . . 20 1.3.1 Binomial distribution with parameter 0 < p < 1 . . . . 20 1.3.2 Poisson distribution with parameter λ > 0 . . . . . . . 21 1.3.3 Geometric distribution with parameter 0 < p < 1 . . . 21 1.3.4 Lebesgue measure and uniform distribution . . . . . . 21 1.3.5 Gaussian distribution on R with mean m ∈ R and variance σ 2 > 0 . . . . . . . . . . . . . . . . . . . . . . 22 1.3.6 Exponential distribution on R with parameter λ > 0 . 22 1.3.7 Poisson’s Theorem . . . . . . . . . . . . . . . . . . . . 24 1.4 A set which is not a Borel set . . . . . . . . . . . . . . . . . . 25 2 Random variables 29 2.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Measurable maps . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Integration 39 3.1 Definition of the expected value . . . . . . . . . . . . . . . . . 39 3.2 Basic properties of the expected value . . . . . . . . . . . . . . 42 3.3 Connections to the Riemann-integral . . . . . . . . . . . . . . 48 3.4 Change of variables in the expected value . . . . . . . . . . . . 49 3.5 Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Some inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Modes of convergence 63 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . 64 3 4 CONTENTS Introduction The modern period of probability theory is connected with names like S.N. Bernstein (1880-1968), E. Borel (1871-1956), and A.N. Kolmogorov (1903- 1987). In particular, in 1933 A.N. Kolmogorov published his modern ap- proach of Probability Theory, including the notion of a measurable space and a probability space. This lecture will start from this notion, to continue with random variables and basic parts of integration theory, and to finish with some first limit theorems. The lecture is based on a mathematical axiomatic approach and is intended for students from mathematics, but also for other students who need more mathematical background for their further studies. We assume that the integration with respect to the Riemann-integral on the real line is known. The approach, we follow, seems to be in the beginning more difficult. But once one has a solid basis, many things will be easier and more transparent later. Let us start with an introducing example leading us to a problem which should motivate our axiomatic approach....
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