A Mathematics Course for Political and Social Research

List of Figures xi List of Tables xii Preface xv I Building Blocks 1 1 Preliminaries 3 1.1 Variables and Constants 3 1.2 Sets 5 1.3 Operators 9 1.4 Relations 13 1.5 Level of Measurement 14 1.6 Notation 18 1.7 Proofs, or How Do We Know This? 22 1.8 Exercises 26 2 Algebra Review 28 2.1 Basic Properties of Arithmetic 28 2.2 Algebra Review 30 2.3 Computational Aids 40 2.4 Exercises 41 3 Functions, Relations, and Utility 44 3.1 Functions 45 3.2 Examples of Functions of One Variable 53 3.3 Preference Relations and Utility Functions 74 3.4 Exercises 78 4 Limits and Continuity, Sequences and Series, and More on Sets 81 4.1 Sequences and Series 81 4.2 Limits 84 4.3 Open, Closed, Compact, and Convex Sets 92 4.4 Continuous Functions 96 4.5 Exercises 99 II Calculus in One Dimension 101 5 Introduction to Calculus and the Derivative 103 5.1 A Brief Introduction to Calculus 103 5.2 What Is the Derivative? 105 5.3 The Derivative, Formally 109 5.4 Summary 114 5.5 Exercises 115 6 The Rules of Differentiation 117 6.1 Rules for Differentiation 118 6.2 Derivatives of Functions 125 6.3 What the Rules Are, and When to Use Them 130 6.4 Exercises 131 7 The Integral 133 7.1 The Defnite Integral as a Limit of Sums 134 7.2 Indefnite Integrals and the Fundamental Theorem of Calculus 136 7.3 Computing Integrals 140 7.4 Rules of Integration 148 7.5 Summary 149 7.6 Exercises 150 8 Extrema in One Dimension 152 8.1 Extrema 153 8.2 Higher-Order Derivatives, Concavity, and Convexity 157 8.3 Finding Extrema 162 8.4 Two Examples 169 8.5 Exercises 170 III Probability 173 9 An Introduction to Probability 175 9.1 Basic Probability Theory 175 9.2 Computing Probabilities 182 9.3 Some Specifc Measures of Probabilities 192 9.4 Exercises 194 9.5 Appendix 197 10 An Introduction to (Discrete) Distributions 198 10.1 The Distribution of a Single Concept (Variable) 199 10.2 Sample Distributions 202 10.3 Empirical Joint and Marginal Distributions 206 10.4 The Probability Mass Function 209 10.5 The Cumulative Distribution Function 216 10.6 Probability Distributions and Statistical Modeling 218 10.7 Expectations of Random Variables 229 10.8 Summary 239 10.9 Exercises 239 10.10 Appendix 241 11 Continuous Distributions 242 11.1 Continuous Random Variables 242 11.2 Expectations of Continuous Random Variables 249 11.3 Important Continuous Distributions for Statistical Modeling 258 11.4 Exercises 271 11.5 Appendix 272 IV Linear Algebra 273 12 Fun with Vectors and Matrices 275 12.1 Scalars 276 12.2 Vectors 277 12.3 Matrices 282 12.4 Properties of Vectors and Matrices 297 12.5 Matrix Illustration of OLS Estimation 298 12.6 Exercises 300 13 Vector Spaces and Systems of Equations 304 13.1 Vector Spaces 305 13.2 Solving Systems of Equations 310 13.3 Why Should I Care? 320 13.4 Exercises 324 13.5 Appendix 326 14 Eigenvalues and Markov Chains 327 14.1 Eigenvalues, Eigenvectors, and Matrix Decomposition 328 14.2 Markov Chains and Stochastic Processes 340 14.3 Exercises 351 V Multivariate Calculus and Optimization 353 15 Multivariate Calculus 355 15.1 Functions of Several Variables 356 15.2 Calculus in Several Dimensions 359 15.3 Concavity and Convexity Redux 371 15.4 Why Should I Care? 372 15.5 Exercises 374 16 Multivariate Optimization 376 16.1 Unconstrained Optimization 377 16.2 Constrained Optimization: Equality Constraints 383 16.3 Constrained Optimization: Inequality Constraints 391 16.4 Exercises 398 17 Comparative Statics and Implicit Differentiation 400 17.1 Properties of the Maximum and Minimum 401 17.2 Implicit Differentiation 405 17.3 Exercises 411 Bibliography 413 Index 423