book

An Introduction to Probability and Statistical Inference, 2nd Edition

Name: An Introduction to Probability and Statistical Inference, 2nd Edition
Author: George G. Roussas
ISBN: 9780128004371

by George G. Roussas

October 2014

Beginner

624 pages

21h 45m

English

Academic Press

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Cover image
Title page
Table of Contents
Inside Front Matter
Copyright
Dedication
Preface
OverviewChapter DescriptionsFeaturesBrief Preface of the Revised VersionAcknowledgments and Credits
Chapter 1: Some motivating examples and some fundamental concepts
1.1 Some Motivating Examples1.2 Some Fundamental Concepts1.3 Random Variables
Chapter 2: The concept of probability and basic results
2.1 Definition of Probability and Some Basic Results2.2 Distribution of a Random Variable2.3 Conditional Probability and Related Results
2.4 Independent Events and Related Results
2.5 Basic Concepts and Results in Counting

Chapter 3: Numerical characteristics of a random variable, some special random variables
3.1 Expectation, Variance, and Moment Generating Function of a Random Variable3.2 Some Probability Inequalities3.3 Some Special Random Variables
3.4 Median and Mode of a Random Variable
Chapter 4: Joint and conditional p.d.f.’s, conditional expectation and variance, moment generating function, covariance, and correlation coefficient
4.1 Joint D.F. and Joint p.d.f. of Two Random Variables4.2 Marginal and Conditional p.d.f.'s, Conditional Expectation and Variance
4.3 Expectation of a Function of Two r.v.'s, Joint and Marginal m.g.f.'s, Covariance, and Correlation Coefficient
4.4 Some Generalizations to k Random Variables
4.5 The Multinomial, the Bivariate Normal, and the Multivariate Normal Distributions
Chapter 5: Independence of random variables and some applications
5.1 Independence of Random Variables and Criteria of Independence5.2 The Reproductive Property of Certain Distributions
Chapter 6: Transformation of random variables
6.1 Transforming a Single Random Variable6.2 Transforming Two or More Random Variables
6.3 Linear Transformations
6.4 The Probability Integral Transform6.5 Order Statistics
Chapter 7: Some modes of convergence of random variables, applications
7.1 Convergence in Distribution or in Probability and Their Relationship7.2 Some Applications of Convergence in Distribution: WLLN and CLT
7.3 Further Limit Theorems
Chapter 8: An overview of statistical inference
8.1 The Basics of Point Estimation8.2 The Basics of Interval Estimation8.3 The Basics of Testing Hypotheses8.4 The Basics of Regression Analysis8.5 The Basics of Analysis of Variance8.6 The Basics of Nonparametric Inference
Chapter 9: Point estimation
9.1 Maximum Likelihood Estimation: Motivation and Examples
9.2 Some Properties of MLE's
9.3 Uniformly Minimum Variance Unbiased Estimates
9.4 Decision-Theoretic Approach to Estimation
9.5 Other Methods of Estimation
Chapter 10: Confidence intervals and confidence regions
10.1 Confidence Intervals10.2 Confidence Intervals in The Presence of Nuisance Parameters10.3 A Confidence Region for (μ, σ2) in the N(μ, σ2) Distribution10.4 Confidence Intervals with Approximate Confidence Coefficient
Chapter 11: Testing hypotheses
11.1 General Concepts, Formulation of Some Testing Hypotheses11.2 Neyman-Pearson Fundamental Lemma, Exponential Type Families, UMP Tests for Some Composite Hypotheses11.3 Some Applications of Theorems 2
11.4 Likelihood Ratio Tests
Chapter 12: More about testing hypotheses
12.1 Likelihood Ratio Tests in the Multinomial Case and Contingency Tables12.2 A Goodness-of-Fit Test12.3 Decision-Theoretic Approach to Testing Hypotheses12.4 Relationship between Testing Hypotheses and Confidence Regions
Chapter 13: A simple linear regression model
13.1 Setting up The Model—The Principle of Least Squares13.2 The Least Squares Estimates of β1 and β2 and Some of Their Properties13.3 Normally Distributed Errors: MLE's of β1, β2, and σ2, Some Distributional Results
13.4 Confidence Intervals and Hypotheses Testing Problems
13.5 Some Prediction Problems13.6 Proof of Theorem 513.7 Concluding Remarks
Chapter 14: Two models of analysis of variance
14.1 One-Way Layout with the Same Number of Observations Per Cell
Chapter 15: Some topics in nonparametric inference
15.1 Some Confidence Intervals with Given Approximate Confidence Coefficient15.2 Confidence Intervals for Quantiles of a Distribution Function15.3 The Two-Sample Sign Test15.4 The Rank Sum and the Wilcoxon–Mann–Whitney Two-Sample Tests
15.5 Nonparametric Curve Estimation
Tables
Some notation and abbreviations
Answers to even-numbered exercises
Chapter 1Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6Chapter 7Chapter 9
Chapter 10
Chapter 11Chapter 12Chapter 13Chapter 14Chapter 15
Index
Inside Back Matter

Content preview from An Introduction to Probability and Statistical Inference, 2nd Edition

6.3 Linear Transformations

In this section, a brief discussion is presented for a specific kind of transformation, linear transformations. The basic concepts and results used here can be found in any textbook on linear algebra.

Definition 3

Suppose the variables x₁,…,x_k are transformed into the variables y₁,…, y_k in the following manner:

$y_{i} = \sum_{j = 1}^{k} c_{i j} x_{j}, ? ? ? i = 1, \dots, k,$

si165_e (14)

where the c_ij's are real constants. Such a transformation is called a linear transformation (all the x_i's enter into the transformation in a linear way, in the first power).

Some terminology and elementary facts from matrix algebra will be used here. Denote by C the k × k

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An Introduction to Probability and Statistical Inference, 2nd Edition

by George G. Roussas

6.3 Linear Transformations

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