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This text is intended primarily for a first course in mathematical probability for students in mathematics, statistics, operations research, engineering, and computer science. It is also appropriate for mathematically oriented students 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.
Features
I. FUNDAMENTALS OF PROBABILITY.
1. Probability Basics.
Biography: Girolamo Cardano.
From Percentages to Probabilities.
Set Theory.
2. Mathematical Probability.
Biography: Andrei Kolmogorov.
Sample Space and Events.
Axioms of Probability.
Specifying Probabilities.
Basic Properties of Probability.
3. Combinatorial Probability.
Biography: James Bernoulli.
The Basic Counting Rule.
Permutations and Combinations.
Applications of Counting Rules to Probability.
4. Conditional Probability and Independence.
Biography: Thomas Bayes.
Conditional Probability.
The General Multiplication Rule.
Independent Events.
Bayes' Rule.
II. DISCRETE RANDOM VARIABLES.
5. Discrete Random Variables and Their Distributions.
Biography: Siméon-Dennis Poisson.
From Variables to Random Variables.
Probability Mass Functions.
Binomial Random Variables.
Hypergeometric Random Variables.
Poisson Random Variables.
Geometric Random Variables.
Other Important Discrete Random Variables.
Functions of a Discrete Random Variable.
6. Jointly Discrete Random Variables.
Biography: Blaise Pascal.
Joint and Marginal Probability Mass Functions: Bivariate Case.
Joint and Marginal Probability Mass Functions: Multivariate Case.
Conditional Probability Mass Functions.
Independent Random Variables.
Functions of Two or More Discrete Random Variables.
Sums of Discrete Random Variables.
7. Expected Value of Discrete Random Variables.
Biography: Christiaan Huygens.
From Averages to Expected Values.
Basic Properties of Expected Value.
Variance of Discrete Random Variables.
Variance, Covariance, and Correlation.
Conditional Expectation.
III. CONTINUOUS RANDOM VARIABLES.
8. Continuous Random Variables and Their Distributions.
Biography: Carl Friedrich Gauss.
Introducing Continuous Random Variables.
Cumulative Distribution Functions.
Probability Density Functions.
Uniform and Exponential Random Variables.
Normal Random Variables.
Other Important Continuous Random Variables.
Functions of a Continuous Random Variable.
9. Jointly Continuous Random Variables.
Biography: Pierre de Fermat.
Joint Cumulative Distribution Functions.
Introducing Joint Probability Density Functions.
Basic Properties of Joint Probability Density Functions.
Marginal and Conditional Probability Density Functions.
Independent Continuous Random Variables.
Functions of Two or More Continuous Random Variables.
Sums and Quotients of Continuous Random Variables.
Multidimensional Transformation Theorem.
10. Expected Value of Continuous Random Variables.
Biography: Pafnuty Chebyshev.
Expected Value of a Continuous Random Variable.
Basic Properties of Expected Value.
Variance, Covariance, and Correlation.
Conditional Expectation.
The Bivariate Normal Distribution.
IV. LIMIT THEOREMS AND ADVANCED TOPICS.
11. Generating Functions and Limit Theorems.
Biography: William Feller.
Moment Generating Functions.
Joint Moment Generating Functions.
Laws of Large Numbers.
The Central Limit Theorem.
12. Additional Topics.
Biography: Sir Ronald Fisher.
The Poisson Process.
Basic Queueing Theory.
The Multivariate Normal Distribution.
Sampling Distributions.
Appendices.
Index.
This text is intended primarily for a first course in mathematical probability for students in mathematics, statistics, operations research, engineering, and computer science. It is also appropriate for mathematically oriented students 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.
Features
Table of Contents
I. FUNDAMENTALS OF PROBABILITY.
1. Probability Basics.
Biography: Girolamo Cardano.
From Percentages to Probabilities.
Set Theory.
2. Mathematical Probability.
Biography: Andrei Kolmogorov.
Sample Space and Events.
Axioms of Probability.
Specifying Probabilities.
Basic Properties of Probability.
3. Combinatorial Probability.
Biography: James Bernoulli.
The Basic Counting Rule.
Permutations and Combinations.
Applications of Counting Rules to Probability.
4. Conditional Probability and Independence.
Biography: Thomas Bayes.
Conditional Probability.
The General Multiplication Rule.
Independent Events.
Bayes' Rule.
II. DISCRETE RANDOM VARIABLES.
5. Discrete Random Variables and Their Distributions.
Biography: Siméon-Dennis Poisson.
From Variables to Random Variables.
Probability Mass Functions.
Binomial Random Variables.
Hypergeometric Random Variables.
Poisson Random Variables.
Geometric Random Variables.
Other Important Discrete Random Variables.
Functions of a Discrete Random Variable.
6. Jointly Discrete Random Variables.
Biography: Blaise Pascal.
Joint and Marginal Probability Mass Functions: Bivariate Case.
Joint and Marginal Probability Mass Functions: Multivariate Case.
Conditional Probability Mass Functions.
Independent Random Variables.
Functions of Two or More Discrete Random Variables.
Sums of Discrete Random Variables.
7. Expected Value of Discrete Random Variables.
Biography: Christiaan Huygens.
From Averages to Expected Values.
Basic Properties of Expected Value.
Variance of Discrete Random Variables.
Variance, Covariance, and Correlation.
Conditional Expectation.
III. CONTINUOUS RANDOM VARIABLES.
8. Continuous Random Variables and Their Distributions.
Biography: Carl Friedrich Gauss.
Introducing Continuous Random Variables.
Cumulative Distribution Functions.
Probability Density Functions.
Uniform and Exponential Random Variables.
Normal Random Variables.
Other Important Continuous Random Variables.
Functions of a Continuous Random Variable.
9. Jointly Continuous Random Variables.
Biography: Pierre de Fermat.
Joint Cumulative Distribution Functions.
Introducing Joint Probability Density Functions.
Basic Properties of Joint Probability Density Functions.
Marginal and Conditional Probability Density Functions.
Independent Continuous Random Variables.
Functions of Two or More Continuous Random Variables.
Sums and Quotients of Continuous Random Variables.
Multidimensional Transformation Theorem.
10. Expected Value of Continuous Random Variables.
Biography: Pafnuty Chebyshev.
Expected Value of a Continuous Random Variable.
Basic Properties of Expected Value.
Variance, Covariance, and Correlation.
Conditional Expectation.
The Bivariate Normal Distribution.
IV. LIMIT THEOREMS AND ADVANCED TOPICS.
11. Generating Functions and Limit Theorems.
Biography: William Feller.
Moment Generating Functions.
Joint Moment Generating Functions.
Laws of Large Numbers.
The Central Limit Theorem.
12. Additional Topics.
Biography: Sir Ronald Fisher.
The Poisson Process.
Basic Queueing Theory.
The Multivariate Normal Distribution.
Sampling Distributions.
Appendices.
Index.