A Course in Probability

by Weiss, Neil A.

• ISBN13: 9780201774719
• ISBN10: 0201774712
• Edition: 1st
• Format: Paperback
• Copyright: 2005-02-08
• Publisher: Pearson

Summary

This text is intended primarily for readers interested in mathematical probability as applied to mathematics, statistics, operations research, engineering, and computer science. It is also appropriate for mathematically oriented readers in the physical and social sciences. Prerequisite material consists of basic set theory and a firm foundation in elementary calculus, including infinite series, partial differentiation, and multiple integration. Some exposure to rudimentary linear algebra (e.g., matrices and determinants) is also desirable. This text includes pedagogical techniques not often found in books at this level, in order to make the learning process smooth, efficient, and enjoyable. Fundamentals of Probability:Probability Basics. Mathematical Probability. Combinatorial Probability. Conditional Probability and Independence.Discrete Random Variables:Discrete Random Variables and Their Distributions. Jointly Discrete Random Variables. Expected Value of Discrete Random Variables.ContinuousRandom Variables:Continuous Random Variables and Their Distributions. Jointly Continuous Random Variables. Expected Value of Continuous Random Variables.Limit Theorems and Advanced Topics:Generating Functions and Limit Theorems. Additional Topics. For all readers interested in probability.

Table of Contents

PART ONE Fundamentals of Probability
Probability Basics
Biography: Girolamo Cardano	2	(1)
Introduction	3	(1)
From Percentages to Probabilities	3	(6)
Set Theory	9	(17)
Chapter Review	20	(4)
Mathematical Probability
Biography: Andrei Kolmogorov	24	(1)
Introduction	25	(1)
Sample Space and Events	26	(13)
Axioms of Probability	39	(10)
Specifying Probabilities	49	(15)
Basic Properties of Probability	64	(22)
Chapter Review	78	(6)
Combinatorial Probability
Biography: Jacob Bernoulli	84	(1)
Introduction	85	(1)
The Basic Counting Rule	86	(8)
Permutations and Combinations	94	(16)
Applications of Counting Rules to Probability	110	(16)
Chapter Review	118	(6)
Conditional Probability and Independence
Biography: Thomas Bayes	124	(1)
Introduction	125	(1)
Conditional Probability	126	(11)
The General Multiplication Rule	137	(10)
Independent Events	147	(13)
Bayes's Rule	160	(16)
Chapter Review	166	(8)
PART TWO Discrete Random Variables
Discrete Random Variables and Their Distributions
Biography: Simeon-Denis Poisson	174	(1)
Introduction	175	(1)
From Variables to Random Variables	176	(8)
Probability Mass Functions	184	(12)
Binomial Random Variables	196	(12)
Hypergeometric Random Variables	208	(11)
Poisson Random Variables	219	(9)
Geometric Random Variables	228	(9)
Other Important Discrete Random Variables	237	(9)
Functions of a Discrete Random Variable	246	(14)
Chapter Review	250	(8)
Jointly Discrete Random Variables
Biography: Blaise Pascal	258	(1)
Introduction	259	(1)
Joint and Marginal Probability Mass Functions: Bivariate Case	260	(12)
Joint and Marginal Probability Mass Functions: Multivariate Case	272	(10)
Conditional Probability Mass Functions	282	(8)
Independent Random Variables	290	(11)
Functions of Two or More Discrete Random Variables	301	(6)
Sums of Discrete Random Variables	307	(19)
Chapter Review	315	(9)
Expected Value of Discrete Random Variables
Biography: Christiaan Huygens	324	(1)
Introduction	325	(1)
From Averages to Expected Values	326	(11)
Basic Properties of Expected Value	337	(15)
Variance of Discrete Random Variables	352	(13)
Variance, Covariance, and Correlation	365	(12)
Conditional Expectation	377	(25)
Chapter Review	391	(9)
PART THREE Continuous Random Variables
Continuous Random Variables and Their Distributions
Biography: Carl Friedrich Gauss	400	(1)
Introduction	401	(1)
Introducing Continuous Random Variables	402	(4)
Cumulative Distribution Functions	406	(10)
Probability Density Functions	416	(12)
Uniform and Exponential Random Variables	428	(9)
Normal Random Variables	437	(13)
Other Important Continuous Random Variables	450	(14)
Functions of a Continuous Random Variable	464	(22)
Chapter Review	476	(8)
Jointly Continuous Random Variables
Biography: Pierre de Fermat	484	(1)
Introduction	485	(1)
Joint Cumulative Distribution Functions	486	(7)
Introducing Joint Probability Density Functions	493	(8)
Properties of Joint Probability Density Functions	501	(8)
Marginal and Conditional Probability Density Functions	509	(14)
Independent Continuous Random Variables	523	(8)
Functions of Two or More Continuous Random Variables	531	(14)
Multivariate Transformation Theorem	545	(19)
Chapter Review	554	(8)
Expected Value of Continuous Random Variables
Biography: Pafnuty Chebyshev	562	(1)
Introduction	563	(1)
Expected Value of a Continuous Random Variable	564	(7)
Basic Properties of Expected Value	571	(11)
Variance, Covariance, and Correlation	582	(13)
Conditional Expectation	595	(12)
The Bivariate Normal Distribution	607	(23)
Chapter Review	619	(9)
PART FOUR Limit Theorems and Applications
Generating Functions and Limit Theorems
Biography: William Feller	628	(1)
Introduction	629	(1)
Moment Generating Functions	630	(11)
Joint Moment Generating Functions	641	(9)
Laws of Large Numbers	650	(9)
The Central Limit Theorem	659	(27)
Chapter Review	675	(9)
Applications of Probability Theory
Biography: Sir Ronald Fisher	684	(1)
Introduction	685	(1)
The Poisson Process	686	(11)
Basic Queueing Theory	697	(15)
The Multivariate Normal Distribution	712	(10)
Sampling Distributions	722
Chapter Review	732
Answers to Selected Exercises	1	(38)
Values of the Standard Normal CDF	39
Index	1