Alan F. Karr

Probability (eBook, PDF)

• Format: PDF

Produktbeschreibung

This book offers a straightforward introduction to the mathematical theory of probability. It presents the central results and techniques of the subject in a complete and self-contained account. As a result, the emphasis is on giving results in simple forms with clear proofs and to eschew more powerful forms of theorems which require technically involved proofs. Throughout there are a wide variety of exercises to illustrate and to develop ideas in the text.

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 283
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461208914
• Artikelnr.: 43991802

Inhaltsangabe

Prelude: Random Walks.- The Model.- Issues and Approaches.- Functional of the Random Walk.- Limit Theorems.- Summary.- 1 Probability.- 1.1 Random Experiments and Sample Spaces.- 1.2 Events and Classes of Sets.- 1.3 Probabilities and Probability Spaces.- 1.4 Probabilities on R.- 1.5 Conditional Probability Given a Set.- 1.6 Complements.- 1.7 Exercises.- 2 Random Variables.- 2.1 Fundamentals.- 2.2 Combining Random Variables.- 2.3 Distributions and Distribution Functions.- 2.4 Key Random Variables and Distributions.- 2.5 Transformation Theory.- 2.6 Random Variables with Prescribed Distributions.- 2.7 Complements.- 2.8 Exercises.- 3 Independence.- 3.1 Independent Random Variables.- 3.2 Functions of Independent Random Variables.- 3.3 Constructing Independent Random Variables.- 3.4 Independent Events.- 3.5 Occupancy Models.- 3.6 Bernoulli and Poisson Processes.- 3.7 Complements.- 3.8 Exercises.- 4 Expectation.- 4.1 Definition and Fundamental Properties.- 4.2 Integrals with respect to Distribution Functions.- 4.3 Computation of Expectations.- 4.4 LP Spaces and Inequalities.- 4.5 Moments.- 4.6 Complements.- 4.7 Exercises.- 5 Convergence of Sequences of Random Variables.- 5.1 Modes of Convergence.- 5.2 Relationships Among the Modes.- 5.3 Convergence under Transformations.- 5.4 Convergence of Random Vectors.- 5.5 Limit Theorems for Bernoulli Summands.- 5.6 Complements.- 5.7 Exercises.- 6 Characteristic Functions.- 6.1 Definition and Basic Properties.- 6.2 Inversion and Uniqueness Theorems.- 6.3 Moments and Taylor Expansions.- 6.4 Continuity Theorems and Applications.- 6.5 Other Transforms.- 6.6 Complements.- 6.7 Exercises.- 7 Classical Limit Theorems.- 7.1 Series of Independent Random Variables.- 7.2 The Strong Law of Large Numbers.- 7.3 The Central Limit Theorem.- 7.4 The Law ofthe Iterated Logarithm.- 7.5 Applications of the Limit Theorems.- 7.6 Complements.- 7.7 Exercises.- 8 Prediction and Conditional Expectation.- 8.1 Prediction in L2.- 8.2 Conditional Expectation Given a Finite Set of Random Variables.- 8.3 Conditional Expectation for X?L2.- 8.4 Positive and Integrable Random Variables.- 8.5 Conditional Distributions.- 8.6 Computational Techniques.- 8.7 Complements.- 8.8 Exercises.- 9 Martingales.- 9.1 Fundamentals.- 9.2 Stopping Times.- 9.3 Optional Sampling Theorems.- 9.4 Martingale Convergence Theorems.- 9.5 Applications of Convergence Theorems.- 9.6 Complements.- 9.7 Exercises.- A Notation.- B Named Objects.