PROBABILITY AND STATISTICS FOR ECONOMISTS BRUCE E. HANSEN Contents Preface x Acknowledgements xi Mathematical Preparation xii Notation xiii 1 Basic Probability Theory 1 1.1 Introduction . 1 1.2 Outcomes and Events . 1 1.3 Probability Function . 3 1.4 Properties of the Probability Function . 4 1.5 Equally-Likely Outcomes . 5 1.6 Joint Events . 5 1.7 Conditional Probability . 6 1.8 Independence . 7 1.9 Law of Total Probability . 9 1.10 Bayes Rule . 10 1.11 Permutations and Combinations . 11 1.12 Sampling With and Without Replacement . 13 1.13 Poker Hands . 14 1.14 Sigma Fields* . 16 1.15 Technical Proofs* . 17 1.16 Exercises . 18 2 Random Variables 22 2.1 Introduction . 22 2.2 Random Variables . 22 2.3 Discrete Random Variables . 22 2.4 Transformations . 24 2.5 Expectation . 25 2.6 Finiteness of Expectations . 26 2.7 Distribution Function . 28 2.8 Continuous Random Variables . 29 2.9 Quantiles . 31 2.10 Density Functions . 31 2.11 Transformations of Continuous Random Variables . 34 2.12 Non-Monotonic Transformations . 36 ii CONTENTS iii 2.13 Expectation of Continuous Random Variables . 37 2.14 Finiteness of Expectations . 39 2.15 Unifying Notation . 39 2.16 Mean and Variance . 40 2.17 Moments . 42 2.18 Jensen’s Inequality . 42 2.19 Applications of Jensen’s Inequality* . 43 2.20 Symmetric Distributions . 45 2.21 Truncated Distributions . 46 2.22 Censored Distributions . 47 2.23 Moment Generating Function . 48 2.24 Cumulants . 50 2.25 Characteristic Function . 51 2.26 Expectation: Mathematical Details* . 52 2.27 Exercises . 52 3 Parametric Distributions 56 3.1 Introduction . 56 3.2 Bernoulli Distribution . 56 3.3 Rademacher Distribution . 57 3.4 Binomial Distribution . 57 3.5 Multinomial Distribution . 58 3.6 Poisson Distribution . 58 3.7 Negative Binomial Distribution . 59 3.8 Uniform Distribution . 59 3.9 Exponential Distribution . 59 3.10 Double Exponential Distribution . 60 3.11 Generalized Exponential Distribution . 60 3.12 Normal Distribution . 61 3.13 Cauchy Distribution . 61 3.14 Student t Distribution . 62 3.15 Logistic Distribution . 62 3.16 Chi-Square Distribution . 63 3.17 Gamma Distribution . 63 3.18 F Distribution . 64 3.19 Non-Central Chi-Square . 64 3.20 Beta Distribution . 65 3.21 Pareto Distribution . 65 3.22 Lognormal Distribution . 66 3.23 Weibull Distribution . 66 3.24 Extreme Value Distribution . 67 3.25 Mixtures of Normals . 68 3.26 Technical Proofs* . 70 3.27 Exercises . 71 4 Multivariate Distributions 74 4.1 Introduction . 74 4.2 Bivariate Random Variables . 75 CONTENTS iv 4.3 Bivariate Distribution Functions . 76 4.4 Probability Mass Function . 78 4.5 Probability Density Function . 79 4.6 Marginal Distribution . 81 4.7 Bivariate Expectation . 82 4.8 Conditional Distribution for Discrete X ............................. 84 4.9 Conditional Distribution for Continuous X ........................... 85 4.10 Visualizing Conditional Densities . 87 4.11 Independence . 89 4.12 Covariance and Correlation . 91 4.13 Cauchy-Schwarz . 93 4.14 Conditional Expectation . ..