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Probability

This book is a text at the introductory graduate level, for use in the one­ semester or two-quarter probability course for first-year graduate students that seems ubiquitous in departments of statistics, biostatistics, mathemat­ ical sciences, applied mathematics and mathematics. While it is accessi...

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Bibliographic Details
Main Author: Karr, Alan F.
Format: eBook
Language:English
Published: New York, NY Springer New York 1993, 1993
Edition:1st ed. 1993
Series:Springer Texts in Statistics
Subjects:
Probability Theory
Probabilities
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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

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