probstat - Probability and Statistics Cheat Sheet Copyright...

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Probability and Statistics Cheat Sheet Copyright c Matthias Vallentin , 2011 [email protected] 6 th March, 2011
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This cheat sheet 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 Univer- sity 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 recent ver- sion of this document is available at http://bit.ly/probstat . 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 Function . . . . . 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) 21 19.2 Non-parametric Regression . . . . . . . 21 19.3 Smoothing Using Orthogonal Functions 21 20 Stochastic Processes 22 20.1 Markov Chains . . . . . . . . . . . . . . 22 20.2 Poisson Processes . . . . . . . . . . . . . 22 21 Time Series 23 21.1 Stationary Time Series . . . . . . . . . . 23 21.2 Estimation of Correlation . . . . . . . . 24 21.3 Non-Stationary Time Series . . . . . . . 24 21.3.1 Detrending . . . . . . . . . . . . 24 21.4 ARIMA models . . . . . . . . . . . . . . 24 21.4.1 Causality and Invertibility . . . . 25 21.5 Spectral Analysis . . . . . . . . . . . . . 25 22 Math 26 22.1 Gamma Function . . . . . . . . . . . . . 26 22.2 Beta Function . . . . . . . . . . . . . . . 26 22.3 Series . . . . . . . . . . . . . . . . . . . 27 22.4 Combinatorics . . . . . . . . . . . . . . 27
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1 Distribution Overview 1.1 Discrete Distributions Notation 1 F X ( x ) f X ( x ) E [ X ] V [ X ] M X ( s ) Uniform Unif { a, . . . , b } 0 x < a b x c- a +1 b - a a x b 1 x > b I ( a < x < b ) b - a + 1 a + b 2 ( b - a + 1) 2 - 1 12 e as - e - ( b +1) s s ( b - a ) Bernoulli Bern ( p ) (1 - p ) 1 - x p x (1 - p ) 1 - x p p (1 - p ) 1 - p + pe s Binomial Bin ( n, p ) I 1 - p ( n - x, x + 1) n x ! p x (1 - p ) n - x np np (1 - p ) (1 - p + pe s ) n Multinomial Mult ( n, p ) n !
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