cookbook-en

# cookbook-en - Probability and Statistics Cookbook Copyright...

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Unformatted text preview: Probability and Statistics Cookbook Copyright c Matthias Vallentin , 2011 [email protected] 19 th July, 2011 This cookbook integrates a variety of topics in probability the- ory and statistics. It is based on literature [ 1 , 6 , 3 ] and in-class material from courses of the statistics department at the Uni- versity of California in Berkeley but also influenced by other sources [ 4 , 5 ]. If you find errors or have suggestions for further topics, I would appreciate if you send me an email . The most re- cent version of this document is available at http://matthias. vallentin.net/probability-and-statistics-cookbook/ . To reproduce, please contact me. Contents 1 Distribution Overview 3 1.1 Discrete Distributions . . . . . . . . . . 3 1.2 Continuous Distributions . . . . . . . . 4 2 Probability Theory 6 3 Random Variables 6 3.1 Transformations . . . . . . . . . . . . . 7 4 Expectation 7 5 Variance 7 6 Inequalities 8 7 Distribution Relationships 8 8 Probability and Moment Generating Functions 9 9 Multivariate Distributions 9 9.1 Standard Bivariate Normal . . . . . . . 9 9.2 Bivariate Normal . . . . . . . . . . . . . 9 9.3 Multivariate Normal . . . . . . . . . . . 9 10 Convergence 9 10.1 Law of Large Numbers (LLN) . . . . . . 10 10.2 Central Limit Theorem (CLT) . . . . . 10 11 Statistical Inference 10 11.1 Point Estimation . . . . . . . . . . . . . 10 11.2 Normal-Based Confidence Interval . . . 11 11.3 Empirical distribution . . . . . . . . . . 11 11.4 Statistical Functionals . . . . . . . . . . 11 12 Parametric Inference 11 12.1 Method of Moments . . . . . . . . . . . 11 12.2 Maximum Likelihood . . . . . . . . . . . 12 12.2.1 Delta Method . . . . . . . . . . . 12 12.3 Multiparameter Models . . . . . . . . . 12 12.3.1 Multiparameter delta method . . 13 12.4 Parametric Bootstrap . . . . . . . . . . 13 13 Hypothesis Testing 13 14 Bayesian Inference 14 14.1 Credible Intervals . . . . . . . . . . . . . 14 14.2 Function of parameters . . . . . . . . . . 14 14.3 Priors . . . . . . . . . . . . . . . . . . . 15 14.3.1 Conjugate Priors . . . . . . . . . 15 14.4 Bayesian Testing . . . . . . . . . . . . . 15 15 Exponential Family 16 16 Sampling Methods 16 16.1 The Bootstrap . . . . . . . . . . . . . . 16 16.1.1 Bootstrap Confidence Intervals . 16 16.2 Rejection Sampling . . . . . . . . . . . . 17 16.3 Importance Sampling . . . . . . . . . . . 17 17 Decision Theory 17 17.1 Risk . . . . . . . . . . . . . . . . . . . . 17 17.2 Admissibility . . . . . . . . . . . . . . . 17 17.3 Bayes Rule . . . . . . . . . . . . . . . . 18 17.4 Minimax Rules . . . . . . . . . . . . . . 18 18 Linear Regression 18 18.1 Simple Linear Regression . . . . . . . . 18 18.2 Prediction . . . . . . . . . . . . . . . . . 19 18.3 Multiple Regression . . . . . . . . . . . 19 18.4 Model Selection . . . . . . . . . . . . . . 19 19 Non-parametric Function Estimation 20 19.1 Density Estimation . . . . . . . . . . . . 20 19.1.1 Histograms . . . . . . . . . . . . 20 19.1.2 Kernel Density Estimator (KDE)19....
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cookbook-en - Probability and Statistics Cookbook Copyright...

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