Fundamentals of Statistics
Preface
1
Introduction
1.1
Tools
1.1.1
R
1.1.2
RStudio
1.1.3
Python
1.1.4
Jupyter
1.1.5
Stan
1.1.6
JASP
1.1.7
Orange
1.1.8
JAMOVI
1.1.9
PSPP
1.1.10
LibreOffice Calc
1.1.11
Kedro
1.1.12
EDF Browser
1.1.13
Tabula
1.1.14
Maple Calculator
1.1.15
Nomograms
1.2
Educational resources
1.2.1
Teacher’s webpage
1.2.2
Khan Academy
1.2.3
Eddie Woo
1.2.4
UFPR
1.2.5
UFSCAR
1.2.6
USP
1.2.7
Seeing theory
1.2.8
Advanced
1.3
Data
1.3.1
Kaggle
1.3.2
UCI Machine Learning Repository
1.3.3
Rdatasets
1.3.4
Base dos Dados
1.3.5
Open Data and National Data Catalog
1.4
Digits and Numbers
1.4.1
Representations
1.5
Date and time
1.6
NA
1.7
Mr.
\(X\)
1.8
Summation
1.9
Bounding the precision
1.10
Other symbols and expressions
1.10.1
Greek alphabet
1.11
Glossary
1.12
Momentinho Cultural
2
Descriptive Statistics
2.1
Variables
2.1.1
Nominal scale
2.1.2
Ordinal scale
2.1.3
Discrete
2.1.4
Continuous
2.1.5
Final remarks
2.2
Frequency Distribution
2.2.1
Raw Data, List/Array and Order Statistics
2.2.2
Discrete frequency distribution
2.2.3
Continuous frequency distribution
2.2.4
Bivariate frequency distribution
2.3
Measures of Location
2.3.1
Minimum and Maximum
2.3.2
Total
2.3.3
(Arithmetic) Mean
2.3.4
Mean Square
2.3.5
Harmonic Mean
2.3.6
Geometric Mean
2.3.7
Relationship between means
2.3.8
Mode
2.3.9
Quantile
2.3.10
5-number summary
2.4
Measures of Dispersion
2.4.1
Range
2.4.2
Variance
2.4.3
Standard deviation
2.4.4
Coefficient of variation
2.4.5
Interquartile Range
2.4.6
Median Absolute Deviation
2.5
Other Measures
2.5.1
Skewness
2.5.2
Kurtosis
2.6
Visualization
2.6.1
Examples
2.6.2
Basic charts
2.6.3
Cookbooks menu
2.6.4
To know more
3
Probability
3.1
Set theory
3.1.1
Empty set
3.1.2
Operations
3.1.3
Power set
3.1.4
Disjoint sets and partition
3.2
Combinatorics
3.2.1
Rule of product
3.2.2
\(r\)
-Arrangement
3.2.3
Permutation
3.2.4
Combination
3.3
Definitions
3.3.1
Random experiment
3.3.2
Sample space
3.3.3
Event
3.3.4
Probability
3.3.5
Fundamental properties (Kolmogorov axioms)
3.3.6
Secondary properties
3.4
Conditional probability
3.4.1
Independence
3.4.2
Conditional independence
3.5
Law of Total Probability and Bayes’ Theorem
3.6
Random Variables
3.6.1
Definition
3.6.2
Probability distribution
3.6.3
Expected value
3.6.4
Variance and standard deviation
3.7
Special Discrete Distributions
3.7.1
Discrete uniform
\(\cdot \; \mathcal{DU}(a,b)\)
3.7.2
Binomial
\(\cdot \; \mathcal{B}(n,p)\)
3.7.3
Negative Binomial
\(\cdot \; \mathcal{BN}(k,p)\)
3.7.4
Poisson
\(\cdot \; \mathcal{P}(\lambda)\)
3.7.5
Hypergeometric
\(\cdot \; \mathcal{H}(N,R,n)\)
3.7.6
Bernoulli-Poisson
\(\cdot \; \mathcal{BP}(p,\lambda_1,\lambda_2)\)
3.8
Continous Random Variables
3.8.1
Expected value
3.8.2
Variance and standard deviation
3.9
Special Continuous Distributions
3.9.1
Continuous Uniform
\(\cdot \; \mathcal{U}(a,b)\)
3.9.2
Exponential
\(\cdot \; \mathcal{E}(\lambda)\)
3.9.3
Normal
\(\cdot \; \mathcal{N}(\mu,\sigma)\)
3.9.4
Chi-squared
\(\cdot \; \mathcal{\chi}^2_\nu\)
3.9.5
Student’s
\(t\)
\(\cdot \; \mathcal{t_\nu}\)
3.9.6
Fisher-Snedecor
\(\cdot \; \mathcal{F}_\nu\)
3.9.7
Beta
\(\cdot \; \mathcal{Beta}(\alpha,\beta)\)
3.9.8
Gamma
\(\cdot \; \mathcal{Gama}(k,g)\)
3.9.9
Triangular
\(\cdot \; \mathcal{Tri}(a,m,b)\)
3.9.10
Gompertz
\(\cdot \; \mathcal{Gompertz}(\alpha,\beta)\)
3.9.11
Unit-Gompertz
\(\cdot \; \mathcal{GU}(\alpha,\beta)\)
3.9.12
Continuous Poisson
3.10
R as table
3.11
Functions of random variables
4
Sampling
4.1
Basic definitions
4.1.1
Elementary unit
4.1.2
Sample unit
4.1.3
Referral system
4.2
Universe or Population
\(\mathcal{U}\)
4.2.1
Parameter
4.3
Samples
4.3.1
Sampling plan
4.3.2
Sampling distributions
4.3.3
Representative sample
4.3.4
Sample types
4.4
Main sampling techniques
4.4.1
Simple Random Sample
4.4.2
Systematic Sampling
4.4.3
Stratified Sampling
4.5
Election polls
4.6
Sample size calculation
4.6.1
Average
4.6.2
Proportion
4.7
To know more
5
Bayesian Inference
5.1
An essay
5.2
Exchangeability
5.3
Likelihood function
5.4
Likelihood principle
5.4.1
See also
5.5
Prior distribution
5.5.1
Jeffreys’ prior
5.5.2
Reference prior
5.5.3
Subjective prior
5.6
Posterior distribution
5.6.1
Conjugate distributions
5.7
Simulation
5.8
Point estimation
5.9
Credibility Interval/Region
5.10
Hypothesis Testing
5.11
To know more
6
Classical Inference
6.1
Point estimation
6.1.1
Unbiased estimators
6.1.2
Maximum likelihood estimators
6.2
Confidence Interval
6.2.1
Proportion (
\(\pi\)
)
6.2.2
Mean (
\(\mu\)
)
6.2.3
Variance (
\(\sigma^2\)
)
6.2.4
Standard deviation (
\(\sigma\)
)
6.3
Hypothesis Testing
6.3.1
Via confidence intervals
6.3.2
Neyman-Pearson lemma
6.3.3
Parametric space
6.3.4
Null hypothesis
\(H_0\)
6.3.5
Alternative hypothesis
\(H_1\)
6.3.6
Test statistic
6.3.7
Error types
6.3.8
p-value
6.3.9
Univariate Parametrics
7
Correlation and Regression
7.1
Correlation
7.1.1
\(\rho\)
\(\cdot\)
Universal correlation
7.1.2
\(r\)
\(\cdot\)
Pearson Correlation Coefficient
7.1.3
\(\rho_{RTO}\)
and
\(r_{RTO}\)
\(\cdot\)
Correlation in RTO
7.1.4
\(r^2\)
\(\cdot\)
Coefficient of determination
7.2
Simple Linear Regression
7.2.1
Model
7.2.2
Inferential diagnostics
7.2.3
Predictive diagnostics
7.3
Regressão Linear Múltipla
7.3.1
Modelo
7.3.2
Diagnóstico inferencial
8
Resolution suggestions
8.1
Chapter 1
8.2
Chapter 2
8.3
Chapter 3
8.4
Chapter 6
9
References
Published with bookdown
Fundamentals of Statistics
5.11
To know more
Nicenboim, Schad and Vasishth (2023-02-18) - An Introduction to Bayesian Data Analysis for Cognitive Science
(
Martin, Kumar, and Lao 2021
)
https://statswithr.github.io/book/
References
Martin, Osvaldo A., Ravin Kumar, and Junpeng Lao. 2021.
Bayesian Modeling and Computation in Python
.
Boca Raton
: CRC Press.
https://bayesiancomputationbook.com/welcome.html
.