Mastering Statistics with R
Welcome
Prerequisites
Who am I ?
Acknowledgement
Progress of this book
Part I: Basic Probability
1
Introduction to Probability
1.1
What is probability ?
1.2
Basic Mathematic
1.2.1
Combinatorics
1.2.2
Set Theory
1.3
History of probability
1.3.1
Experiment, Sample space and Events
1.3.2
Definitions of Probability
1.4
Conditional Probability and Independence
1.4.1
Conditional Probability
1.4.2
Independence
1.5
Bayes’ Theorem
2
Random variables
2.1
Random variables and probability functions
2.1.1
Random variables
2.1.2
Discrete Probability Function
2.1.3
Continuous Probability Function
2.1.4
* Mixed Type Probability Function
2.2
Expected values and Variance
2.2.1
*Approximation of a random variable
2.3
Transformation of random variables
2.3.1
Discrete r.v. transformation
2.3.2
Continuous r.v. transformation
2.4
Families of distributions
2.4.1
Discrete probability distributions
2.4.2
Continuous probability distributions
3
Multivariate random variables
3.1
Joint distributions
3.1.1
Marginal Distribution
3.1.2
Sum of two independent random variables
3.2
Change of variables
3.3
Families of multivariate distributions
3.3.1
Trinomial distribution
3.3.2
Bivariate hypergeometric distribution
3.3.3
Multivariate normal distribution
3.3.4
Wishart distribution
3.3.5
Wilks’ lambda distribution
3.3.6
Hotelling’s
\(T^2\)
-distribution
4
System of Moments
5
Limit Theorem
5.1
Some inequality
5.1.1
Markov inequality
5.1.2
Chebshev inequality
5.1.3
Jensen inequality
5.2
Law of large numbers
Part II: Basic Statistic
6
Descriptive Statistics
7
Sampling
8
Estimation
8.1
Point Estimation
8.1.1
Method of Moments (MoM)
8.1.2
Maximum Likelihood Estimation (MLE)
8.1.3
Uniformly Minimum Variance Unbiased Estimator (UMVUE)
8.2
Interval Estimation
9
Testing Hypotheses
9.1
Null hypothesis vs. alternative hypothesis
9.2
The Neyman-Pearson Lemma
10
Some statistical test
10.1
Parametric statistical test
10.1.1
\(t\)
-test
10.1.2
\(F\)
-test
10.1.3
\(\chi^2\)
-test
10.2
Non-parametric statistical test
10.2.1
Mann–Whitney
\(U\)
-test (Wilcoxon rank-sum test)
10.2.2
Wilcoxon signed-rank test
10.2.3
Kolmogorov–Smirnov test
11
Analysis of Variance (ANOVA)
11.1
One-way ANOVA
11.2
Two-way ANOVA
11.3
Kruskal–Wallis test
12
Correlation Analysis and Linear Regression
13
Limiting Distributions
13.1
Converge in probability
13.2
Converge in distribution
Part III: Some Statistic Topic
14
(Generalized) Linear Models
15
Applied Probability Models
16
Time Series Analysis
17
Statistical Learning
18
Statistical Computing
18.1
Generate random variables
18.1.1
Inverse transform method
18.1.2
Accept-Rejection method
18.2
Variance reduction
18.3
Monte-Carlo and Markov chain (MCMC)
19
Multivariate Analysis
20
Survial Analysis
21
Categorical Data Analysis
22
Biostatistical Data Analysis
23
Quality Control
24
Causal Inference
25
Statistical Designs and Analyses in Clinical Trials
26
Consulting in Statistics
27
Probability Theory
27.1
Basics from Measure Theory
27.2
Limit of the sets
28
Spatial Statistics
29
Bayesian Analysis
Part IV: Deal with Computer Science
30
Data Visualization and Visual Analytics
31
Big Data Analytics Techniques and Applications
32
Data Mining
33
Deep Learning
Appendix
A
Matrix calculus
B
Functions in R
Published with bookdown
Mastering Statistics with R
Chapter 27
Probability Theory
(include real analysis)