Advanced Calculus with Applications in Statistics [Hardcover]

Designed to help motivate the learning of advanced calculus by demonstrating its relevance in the field of statistics, this successful text features detailed coverage of optimization techniques and their applications in statistics while introducing the reader to approximation theory. The Second Edition provides substantial new coverage of the material, including three new chapters and a large appendix that contains solutions to almost all of the exercises in the book. Applications of some of these methods in statistics are discusses. 

Preface xv

Preface to the First Edition xvii

1. An Introduction to Set Theory 1

1.1. The Concept of a Set 1

1.2. Set Operations 2

1.3. Relations and Functions 4

1.4. Finite Countable and Uncountable Sets 6

1.5. Bounded Sets 9

1.6. Some Basic Topological Concepts 10

1.7. Examples in Probability and Statistics 13

Further Reading and Annotated Bibliography 15

Exercises 17

2. Basic Concepts in Linear Algebra 21

2.1. Vector Spaces and Subspaces 21

2.2. Linear Transformations 25

2.3. Matrices and Determinants 27

2.3.1. Basic Operations on Matrices 28

2.3.2. The Rank of a Matrix 33

2.3.3. The Inverse of a Matrix 34

2.3.4. Generalized Inverse of a Matrix 36

2.3.5. Eigenvalues and Eigenvectors of a Matrix 36

2.3.6. Some Special Matrices 38

2.3.7. The Diagonalization of a Matrix 38

2.3.8. Quadratic Forms 39

2.3.9. The Simultaneous Diagonalization of Matrices 40

2.3.10. Bounds on Eigenvalues 41

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