The objective of this repository is to write my numerical notes and observations about the topics I'm reviewing of Probability and Statistics. For me, this book is a great source of knowledge and instruction to the basics principles of probability and a good introduction to advanced statistics.
- 1.1 The History of Probability
- 1.2 Interpretations of Probability
- 1.3 Experiments and Events
- 1.4 Set Theory
- 1.5 The Definition of Probability
- 1.6 Finite Sample Spaces
- 1.7 Counting Methods
- 1.8 Combinatorial Methods
- 1.9 Multinomial Coefficients
- 1.10 The Probability of a Union of Events
- 1.11 Statistical Swindles
- 1.12 Supplementary Exercises
- 2.1 The Definition of Conditional Probability
- 2.2 Independent Events
- 2.3 Bayes’ Theorem
- 2.4 The Gambler's Ruin Problem
- 2.5 Supplementary Exercises
- 3.1 Random Variables and Discrete Distributions
- 3.2 Continuous Distributions
- 3.3 The Cumulative Distribution Function
- 3.4 Bivariate Distributions
- 3.5 Marginal Distributions
- 3.6 Conditional Distributions
- 3.7 Multivariate Distributions
- 3.8 Functions of a Random Variable
- 3.9 Functions of Two or More Random Variables
- 3.10 Markov Chains
- 3.11 Supplementary Exercises
- 4.1 The Expectation of a Random Variable
- 4.2 Properties of Expectations
- 4.3 Variance
- 4.4 Moments
- 4.5 The Mean and the Median
- 4.6 Covariance and Correlation
- 4.7 Conditional Expectation
- 4.8 Utility
- 4.9 Supplementary Exercises
- 5.1 Introduction
- 5.2 The Bernoulli and Binomial Distributions
- 5.3 The Hypergeometric Distributions
- 5.4 The Poisson Distributions
- 5.5 The Negative Binomial Distributions
- 5.6 The Normal Distributions
- 5.7 The Gamma Distributions
- 5.8 The Beta Distributions
- 5.9 The Multinomial Distributions
- 5.10 The Bivariate Normal Distributions
- 5.11 Supplementary Exercises
- 6.1 Introduction
- 6.2 The Law of Large Numbers
- 6.3 The Central Limit Theorem
- 6.4 The Correction for Continuity
- 6.5 Supplementary Exercises
- 7.1 Statistical Inference
- 7.2 Prior and Posterior Distributions
- 7.3 Conjugate Prior Distributions
- 7.4 Bayes Estimates
- 7.5 Maximum Likelihood Estimators
- 7.6 Properties of Maximum Likelihood Estimators
- 7.7 Sufficient Statistics
- 7.8 Jointly Sufficient Statistics
- 7.9 Improving an Estimator
- 7.10 Supplementary Exercises
- 8.1 The Sampling Distribution of a Statistic
- 8.2 The Chi-Square Distributions
- 8.3 Joint Distribution of the Sample Mean and Sample Variance
- 8.4 The t Distributions
- 8.5 Confidence Intervals
- 8.6 Bayesian Analysis of Samples from a Normal Distribution
- 8.7 Unbiased Estimators
- 8.8 Fisher Information
- 8.9 Supplementary Exercises
- 9.1 Problems of Testing Hypotheses
- 9.2 Testing Simple Hypotheses
- 9.3 Uniformly Most Powerful Tests
- 9.4 Two-Sided Alternatives
- 9.5 The T Test
- 9.6 Comparing the Means of Two Normal Distributions
- 9.7 The F Distributions
- 9.8 Bayes Test Procedures
- 9.9 Foundational Issues
- 9.10 Supplementary Exercises
- 10.1 Tests of Goodness-of-Fit
- 10.2 Goodness-of-Fit for Composite Hypotheses
- 10.3 Contingency Tables
- 10.4 Tests of Homogeneity
- 10.5 Simpson's Paradox
- 10.6 Kolmogorov-Smirnov Tests
- 10.7 Robust Estimation
- 10.8 Sign and Rank Tests
- 10.9 Supplementary Exercises
- 11.1 The Method of Least Squares
- 11.2 Regression
- 11.3 Statistical Inference in Simple Linear Regression
- 11.4 Bayesian Inference in Simple Linear Regression
- 11.5 The General Linear Model and Multiple Regression
- 11.6 Analysis of Variance
- 11.7 The Two-Way Layout
- 11.8 The Two-Way Layout with Replications
- 11.9 Supplementary Exercises
- 12.1 What is Simulation?
- 12.2 Why is Simulation Useful?
- 12.3 Simulating Specific Distributions
- 12.4 Importance Sampling
- 12.5 Markov Chain Monte Carlo
- 12.6 The Bootstrap
- 12.7 Supplementary Exercises