Probability for
Data Science
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Chapter 1
Mathematical Background
1.1
Infinite Series
1.2
Approximation
1.3
Integration
1.4
Linear Algebra
1.5
Basic Combinatorics
Summary
Chapter 2
Probability
2.1
Set Theory
2.2
Probability Space
2.3
Axioms of Probability
2.4
Conditional Probability
Summary
Chapter 3
Discrete Random Variables
3.1
Random Variables
3.2
Probability Mass Function
3.3
Cumulative Distribution Functions (Discrete)
3.4
Expectation
3.5
Common Discrete Random Variables
Summary
Chapter 4
Continuous Random Variables
4.1
Probability Density Function
4.2
Expectation, Moment, and Variance
4.3
Cumulative Distribution Function
4.4
Median, Mode, and Mean
4.5
Uniform and Exponential Random Variables
4.6
Gaussian Random Variables
4.7
Functions of Random Variables
4.8
Generating Random Numbers
Summary
Chapter 5
Joint Distributions
5.1
Joint PMF and Joint PDF
5.2
Joint Expectation
5.3
Conditional PMF and PDF
5.4
Conditional Expectation
5.5
Sum of Two Random Variables
5.6
Random Vectors and Covariance Matrices
5.7
Transformation of Multidimensional Gaussians
5.8
Principal-Component Analysis
Summary
Chapter 6
Sample Statistics
6.1
Moment-Generating and Characteristic Functions
6.2
Probability Inequalities
6.3
Law of Large Numbers
6.4
Central Limit Theorem
Summary
Chapter 7
Regression
7.1
Principles of Regression
7.2
Overfitting
7.3
Bias and Variance Trade-Off
7.4
Regularization
Summary
Chapter 8
Estimation
8.1
Maximum-Likelihood Estimation
8.2
Properties of ML Estimates
8.3
Maximum A Posteriori Estimation
8.4
Minimum Mean-Square Estimation
Summary
Chapter 9
Confidence and Hypothesis
9.1
Confidence Interval
9.2
Bootstrapping
9.3
Hypothesis Testing
9.4
Neyman-Pearson Test
9.5
ROC and Precision-Recall Curve
Summary
Chapter 10
Random Processes
10.1
Basic Concepts
10.2
Mean and Correlation Functions
10.3
Wide-Sense Stationary Processes
10.4
Power Spectral Density
10.5
WSS Process through LTI Systems
10.6
Optimal Linear Filter
Summary
Appendix