Probability with Statistical Applications

✍ Scribed by Rinaldo B. Schinazi

• Publisher: Birkhäuser, Cham
• Year: 2022
• Tongue: English
• Leaves: 354
• Edition: 3
• Category: Library

✦ Synopsis

This textbook, now in its third edition, offers a practical introduction to probability with statistical applications, covering material for both a first and second undergraduate probability course. The author focuses on essential concepts that every student should thoroughly understand. The content is organized into brief, easy-to-follow chapters, motivated by plenty of examples.

The first part of the book focuses on classical discrete probability distributions, then goes on to study continuous distributions, confidence intervals, and statistical tests. The following section introduces more advanced concepts suitable for a second course in probability, such as random vectors and sums of random variables. The last part of the book is dedicated to mathematical statistics concepts such as estimation, sufficiency, Bayes' estimation, and multiple regression. This third edition includes a new chapter on combinatorics and a more distinct separation between discrete and continuous distributions. Some of the longer chapters in the previous editions have been divided into shorter chapters to allow for more flexible teaching.

Probability with Statistical Applications, Third Edition is intended for undergraduate students taking a first course in probability; later chapters are also suited for a second course in probability and mathematical statistics. Calculus is the only prerequisite; prior knowledge of probability is not required.

✦ Table of Contents

Preface to the Third Edition
Contents
1 Probability Space
1 Equally Likely Outcomes
2 The Axioms of Probability
Problems
2 Conditional Probabilities
1 Definition
2 Bayes' Method
3 Symmetry
Problems
4 Independence
Problems
5 The Birthday Problem
Problems
3 Discrete Random Variables
1 Discrete Distributions
1.1 Bernoulli Random Variables
1.2 Geometric Random Variables
Problems
2 Expectation
2.1 The Expectation of a Sum
Problems
3 Variance
3.1 Variance and Independence
Problems
4 Coupon Collector's Problem
Problems
4 Binomial Random Variables
1 Binomial Probability Distribution
Problems
2 Mean and Variance
2.1 Derivation of the Binomial Distribution
3 Normal Approximation
3.1 The Normal Table
3.2 Normal Approximation
Problems
4 The Negative Binomial
Problems
5 Poisson Random Variables
1 Poisson Probability Distribution
2 Poisson Scatter Theorem
3 Poisson Approximation to the Binomial
4 Approximation to a Sum of Binomials
Problems
5 Mean and Variance
6 Simulations of Discrete Random Variables
1 Random Numbers
2 Bernoulli Random Variables
3 Binomial Random Variables
3.1 Computational Formula for the Binomial Distribution
4 Poisson Random Variables
Problems
7 Combinatorics
1 Counting Principle
Problems
2 Properties of the Binomial Coefficients
Problems
3 Hypergeometric Random Variables
Problems
4 Mean and Variance of a Hypergeometric
5 Conditioning on the Number of Successes
8 Continuous Random Variables
1 Probability Densities
2 Uniform Random Variables
3 Exponential Random Variables
3.1 Memoryless Property
Problems
4 Expected Value
4.1 Symmetric Probability Density
4.2 Function of a Random Variable
5 The Median
Problems
6 Variance
Problems
7 Normal Random Variables
7.1 The Standard Normal
7.2 Normal Random Variables
7.3 Applications of Normal Random Variables
7.4 Expectation and Variance of a Standard Normal
Problems
More Problems for Chap.8
9 The Sample Average and Variance
1 The Sample Average
2 The Central Limit Theorem
3 The Sample Variance
Problems
4 Monte Carlo Integration
Problem
10 Estimating and Testing Proportions
1 Testing a Proportion
2 Confidence Interval for a Proportion
Problems
3 Testing Two Proportions
4 Confidence Interval for Two Proportions
Problems
11 Estimating and Testing Means
1 Testing a Mean
2 Confidence Interval for a Mean
3 Testing Two Means
4 Two Means Confidence Interval
Problems
12 Small Samples
1 Student Tests
2 Two Means Student Tests
3 Student Tests for Matched Pairs
4 The Sign Test
Problems
13 Chi-Squared Tests
1 Testing Independence
2 Goodness of Fit Test
Problems
14 Design of Experiments
1 Double Blind Design
2 Data Dredging
Problems
15 The Cumulative Distribution Function
1 Definition and Examples
2 Transformations of Random Variables
Problems
3 Sample Maximum and Minimum
Problems
4 Simulations
Problems
16 Continuous Joint Distributions
1 Joint and Marginal Densities
2 Independence
3 Transformations of Random Vectors
Problems
4 Gamma and Beta Random Variables
4.1 The Function Gamma
4.2 Gamma Random Variables
4.3 The Ratio of Two Gamma Random Variables
4.4 Beta Random Variables
Problems
17 Covariance and Independence
1 Covariance
2 Independence
3 Correlation
4 Variance of a Sum
Problems
5 Proof That the Expectation Is Linear
6 Proof That the Correlation Is Bounded
18 Conditional Distribution and Expectation
1 The Discrete Case
Problems
2 Continuous Case
Problems
3 Conditional Expectation
3.1 Conditional Expectation and Prediction
Problems
19 The Bivariate Normal Distribution
1 The Correlation
2 An Application
2.1 Best Predictor
Problems
3 The Joint Probability Density
4 The Conditional Probability Density
Problems
20 Sums of Bernoulli Random Variables
1 The Expected Number of Birthdays
2 The Matching Problem
2.1 Expected Number of Matches
2.2 Variance of a Sum
2.3 Variance of the Number of Matches
3 The Moments of the Hypergeometric
4 The Number of Records
Problems
21 Coupling Random Variables
1 Coupling Two Bernoulli Random Variables
2 Coupling Two Poisson Random Variables
3 The Coupling Inequality
4 Poisson Approximation of a Sum
4.1 Poisson Approximation of a Binomial
5 Proof of the Poisson Approximation
Problems
22 The Moment Generating Function
1 Definition and Examples
1.1 Sum of i.i.d. Bernoulli Random Variables
1.2 Sum of Independent Poisson Random Variables
Problems
2 The m.g.f. of a Normal
3 Moment Computations
Problems
4 Convergence in Distribution
4.1 Binomial Convergence to a Poisson
4.2 Proof of the Central Limit Theorem
Problems
23 Chi-Squared, Student, and F Distributions
1 The m.g.f. of a Gamma Random Variable
1.1 Sum of i.i.d. Exponential Random Variables
Problems
2 The Chi-Squared Distribution
3 The Student Distribution
4 The F Distribution
Problems
24 Sampling from a Normal Distribution
1 The Sample Average and Variance
Problems
2 The Sample Average Is Normal
3 The Sample Variance Distribution
4 The Standardized Average
Problem
25 Finding Estimators
1 The Method of Moments
Problems
2 The Maximum Likelihood Method
Problems
26 Comparing Estimators
1 The Mean Squared Error
2 Biased and Unbiased Estimators
3 Two Estimators for a Normal Variance
4 Two Estimators for a Uniform Distribution
5 Proof of the M.S.E. Formula
Problems
27 Best Unbiased Estimators
1 Exponential Families of Distributions
2 Minimum Variance Unbiased Estimators
Problems
3 Sufficient Statistics
4 A Factorization Theorem
5 Conditional Expectation and Sufficiency
Problems
28 Bayes' Estimators
1 The Prior and Posterior Distributions
2 Bayes' Estimators
Problems
29 Multiple Linear Regression
1 The Least Squares Estimate
2 Statistical Tests
2.1 Sums of Squares
2.2 The R Statistic
2.3 Significance of the Model
2.4 Estimating the Variance
2.5 Testing Individual Regression Coefficients
Problems
3 Proofs
3.1 The Normal Equations
3.2 Partitioning the Sum of Squares
3.3 Expectation and Variance of a Random Vector
3.4 Normal Random Vectors
Problems
List of Common Discrete Distributions
List of Common Continuous Distributions
Further Reading
Probability
Statistics
Standard Normal Table
Student Table
Chi-Squared Table
Index