Probability and Statistics for Data Science : Math + R + Data (Paperback)

Basic Probability Models Example: Bus Ridership A \Notebook" View: the Notion of a Repeatable Experiment Theoretical Approaches A More Intuitive Approach Our Definitions "Mailing Tubes" Example: Bus Ridership Model (cont'd) Example: ALOHA Network ALOHA Network Model Summary ALOHA Network Computations ALOHA in the Notebook Context Example: A Simple Board Game Bayes' Rule General Principle Example: Document Classification Random Graph Models Example: Preferential Attachment Model Combinatorics-Based Probability Computation Which Is More Likely in Five Cards, One King or Two Hearts? Example: Random Groups of Students Example: Lottery Tickets Example: \Association Rules" Example: Gaps between Numbers Multinomial Coefficients Example: Probability of Getting Four Aces in a Bridge Hand Monte Carlo Simulation Example: Rolling Dice First Improvement Second Improvement Third Improvement Example: Dice Problem Use of runif() for Simulating Events Example: ALOHA Network (cont'd) Example: Bus Ridership (cont'd) Example: Board Game (cont'd) Example: Broken Rod How Long Should We Run the Simulation? Computational Complements More on the replicate() Function Discrete Random Variables: Expected Value Random Variables Discrete Random Variables Independent Random Variables Example: The Monty Hall Problem Expected Value Generality|Not Just for Discrete Random Variables Misnomer Definition and Notebook View Properties of Expected Value Computational Formula Further Properties of Expected Value Finding Approximate Expected Values via Simulation Casinos, Insurance Companies and \Sum Users," Compared to Others Mathematical Complements Proof of Property E: Discrete Random Variables: Variance Variance Definition Central Importance of the Concept of Variance Intuition Regarding the Size of Var(X) Chebychev's Inequality The Coefficient of Variation A Useful Fact Covariance Indicator Random Variables, and Their Means and Variances Example: Return Time for Library Books, Version I Example: Return Time for Library Books, Version II Example: Indicator Variables in a Committee Problem Skewness Mathematical Complements Proof of Chebychev's Inequality Discrete Parametric Distribution Families Distributions Example: Toss Coin Until First Head Example: Sum of Two Dice Example: Watts-Strogatz Random Graph Model The Model Parametric Families of Distributions The Case of Importance to Us: Parameteric Families of pmfs Distributions Based on Bernoulli Trials The Geometric Family of Distributions R Functions Example: a Parking Space Problem The Binomial Family of Distributions R Functions Example: Parking Space Model The Negative Binomial Family of Distributions R Functions Example: Backup Batteries Two Major Non-Bernoulli Models The Poisson Family of Distributions R Functions Example: Broken Rod Fitting the Poisson and Power Law Models to Data Example: the Bus Ridership Problem Example: Flipping Coins with Bonuses Example: Analysis of Social Networks Mathematical Complements Computational Complements Graphics and Visualization in R Introduction to Discrete Markov Chains Matrix Formulation Example: Die Game Long-Run State Probabilities Stationary Distribution Calculation of _ Simulation Calculation of _ Example: -Heads-in-a-Row Game Example: Bus Ridership Problem Hidden Markov Models Example: Bus Ridership Computation Google PageRank Continuous Probability Models A Random Dart Individual Values Now Have Probability Zero But Now We Have a Problem Our Way Out of the Problem: Cumulative Distribution Functions CDFs Non-Discrete, Non-Continuous Distributions Density Functions Properties of Densities Intuitive Meaning of Densities Expected Values A First Example Famous Parametric Families of Continuous Distributions The Uniform Distributions Density and Properties R Functions Example: Modeling of Disk Performance Example: Modeling of Denial-of-Service Attack The Normal (Gaussian) Family of Continuous Distributions Density and Properties R Functions Importance in Modeling The Exponential Family of Distributions Density and Properties R Functions Example: Garage Parking Fees Memoryless Property of Exponential Distributions Importance in Modeling The Gamma Family of Distributions Density and Properties Example: Network Buffer Importance in Modeling The Beta Family of Distributions Density Etc Importance in Modeling Mathematical Complements Duality of the Exponential Family with the Poisson Family Computational Complements Inverse Method for Sampling from a Density Sampling from a Poisson Distribution Statistics: Prologue Importance of This Chapter Sampling Distributions Random Samples The Sample Mean | a Random Variable Toy Population Example Expected Value and Variance of X Toy Population Example Again Interpretation Notebook View Simple Random Sample Case The Sample Variance|Another Random Variable Intuitive Estimation of _ Easier Computation Special Case: X Is an Indicator Variable To Divide by n or n-? Statistical Bias The Concept of a \Standard Error" Example: Pima Diabetes Study Don't Forget: Sample = Population! Simulation Issues Sample Estimates Infinite Populations? Observational Studies The Bayesian Philosophy How Does It Work? Arguments for and Against Computational Complements R's split() and tapply() Functions Fitting Continuous Models Estimating a Density from Sample Data Example: BMI Data The Number of Bins The Bias-Variance Tradeo_ The Bias-Variance Tradeo_ in the Histogram Case A General Issue: Choosing the Degree of Smoothing Parameter Estimation Method of Moments Example: BMI Data The Method of Maximum Likelihood Example: Humidity Data MM vs MLE Advanced Methods for Density Estimation Assessment of Goodness of Fit Mathematical Complements Details of Kernel Density Estimators Computational Complements Generic Functions The gmm Package The gmm() Function Example: Bodyfat Data The Family of Normal Distributions Density and Properties Closure Under Affine Transformation Closure Under Independent Summation A Mystery R Functions The Standard Normal Distribution Evaluating Normal cdfs Example: Network Intrusion Example: Class Enrollment Size The Central Limit Theorem Example: Cumulative Roundo_ Error Example: Coin Tosses Example: Museum Demonstration A Bit of Insight into the Mystery X Is Approximately Normal|No Matter What the Population Distribution Is Approximate Distribution of (Centered and Scaled) X Improved Assessment of Accuracy of X Importance in Modeling The Chi-Squared Family of Distributions Density and Properties Example: Error in Pin Placement Importance in Modeling Relation to Gamma Family Mathematical Complements Convergence in Distribution, and the Precisely-Stated CLT Computational Complements Example: Generating Normal Random Numbers Introduction to Statistical Inference The Role of Normal Distributions Confidence Intervals for Means Basic Formulation Example: Pima Diabetes Study Example: Humidity Data Meaning of Confidence Intervals A Weight Survey in Davis Confidence Intervals for Proportions Example: Machine Classification of Forest Covers The Student-t Distribution Introduction to Significance Tests The Proverbial Fair Coin The Basics General Testing Based on Normally Distributed Estimators The Notion of \p-Values" What's Random and What Is Not Example: the Forest Cover Data Problems with Significance Testing History of Significance Testing, and Where We Are Today The Basic Issues Alternative Approach The Problem of \P-hacking" A Thought Experiment Multiple Inference Methods Philosophy of Statistics More about Interpretation of CIs The Bayesian View of Confidence Intervals Multivariate Distributions Multivariate Distributions: Discrete Case Example: Marbles in a Bag Multivariate pmfs Multivariate Distributions: Continuous Case Multivariate Densities Motivation and Definition Use of Multivariate Densities in Finding Probabilities and Expected Values Example: a Triangular Distribution Example: Train Rendezvous Multivariate Distributions: Mixed Discrete-Continuous Case Measuring Co-variation of Random Variables Covariance Example: the Committee Example Again Correlation Example: Correlation in the Triangular Distribution Sample Estimates Sets of Independent Random Variables Properties Expected Values Factor Covariance Is Variances Add Examples Involving Sets of Independent Random Variables Example: Dice Matrix Formulations Properties of Mean Vectors Covariance Matrices Covariance Matrices Linear Combinations of Random Vectors More on Sets of Independent Random Variables Probability Mass Functions and Densities Factor in the Independent Case Convolution Example: Ethernet Example: Backup Battery The Multivariate Normal Family of Distributions Densities Geometric Interpretation R Functions Special Case: New Variable Is a Single Linear Combination of a Random Vector Properties of Multivariate Normal Distributions The Multivariate Central Limit Theorem Iterated Expectations Conditional Distributions The Theorem Example: Flipping Coins with Bonuses Conditional Expectation as a Random Variable What about Variance? Mixture Distributions Derivation of Mean and Variance Mathematical Complements Transform Methods Generating Functions Sums of Independent Poisson Random Variables Are Poisson Distributed A Geometric View of Conditional Expectation Alternate Proof of E(UV) = EU EV for Independent U,V Computational Complements Generating Multivariate Normal Random Vectors Dimension Reduction Principal Components Analysis Intuition Properties of PCA Example: Turkish Teaching Evaluations Mathematical Complements Derivation of PCA Predictive Modeling Example: Heritage Health Prize The Goals: Prediction and Description Terminology What Does \Relationship" Really Mean? Precise Definition Parametric Models for the Regression Function m() Estimation in Linear Parametric Regression Models Example: Baseball Data R Code Multiple Regression: More Than One Predictor Variable Example: Baseball Data (cont'd) Interaction Terms Parametric Estimation of Linear Regression Functions Meaning of \Linear" Random-X and Fixed-X Regression Point Estimates and Matrix Formulation Approximate Confidence Intervals Example: Baseball Data (cont'd) Dummy Variables Classification Classification = Regression Logistic Regression The Logistic Model: Motivations Estimation and Inference for Logit Coefficients Example: Forest Cover Data R Code Analysis of the Results Multiclass Case Machine Learning Methods: Neural Networks Example: Predicting Vertebral Abnormalities But What Is Really Going On? R Packages Mathematical Complements Matrix Derivatives and Minimizing the Sum of Squares Computational Complements Some Computational Details in Section More Regarding glm() Model Parsimony and Overfitting What Is Overfitting? Example: Histograms Example: Polynomial Regression Can Anything Be Done about It? Cross-Validation A. R Quick Start A Correspondences A Starting R A First Sample Programming Session A Vectorization A Second Sample Programming Session A Recycling A More on Vectorization A Third Sample Programming Session A Default Argument Values A The R List Type A The Basics A S Classes A Some Workhorse Functions A Data Frames A Online Help A Debugging in R A Further Reading B. Matrix Algebra B Terminology and Notation B Matrix Addition and Multiplication B Matrix Transpose B Linear Independence B Determinants B Matrix Inverse B Eigenvalues and Eigenvectors B Matrix Algebra in R