Contents
Preface
1
Fundamentals of Probability and its Axioms
1.1
Combinations and Permutations
1.1.1
Permutations
1.1.2
Combinations
1.1.3
Worked Examples
1.2
Review of Set Theory
1.2.1
Laws of Set Theory
1.3
Axioms of Probability
1.3.1
Sample spaces with equally likely outcomes
1.3.2
Worked Examples
2
Conditional Probability and Independence
2.1
Conditional Probability
2.2
The Multiplication Rule
2.3
Independence of Events
2.4
The Law of Total Probability
2.5
Bayes’ Theorem
2.6
Applications
3
Discrete Random Variables and Distributions
3.1
Random Variables
3.2
Probability Mass Functions (PMFs)
3.3
Cumulative Distribution Functions (CDFs)
3.4
Expected Value and Properties
3.5
Variance and Standard Deviation
3.6
Common Discrete Distributions: Binomial, Geometric, Poisson, Negative Binomial, Multinomial
4
Continuous Random Variables and Distributions
4.1
Continuous Random Variables
4.2
Probability Density Functions (PDFs)
4.3
Cumulative Distribution Functions (CDFs)
4.4
Expected Value and Variance (Continuous Case)
4.5
Uniform, Exponential, and Normal Distributions
4.6
Change of Variables and Applications
5
Joint Distributions and Independence
5.1
Joint Discrete Distributions
5.2
Joint Continuous Distributions
5.3
Marginal and Conditional Distributions
5.4
Independence of Random Variables
5.5
Covariance and Correlation
5.6
Sums of Random Variables
6
Limit Theorems
6.1
Sequences of Random Variables
6.2
Convergence in Probability and Distribution
6.3
The Law of Large Numbers (LLN)
6.4
Central Limit Theorem (CLT)
6.5
Applications of the CLT
6.6
Approximations Using the Normal Distribution
Appendix
Algebra Review
Calculus Review
Bibliography
Introduction to Probability
Chapter 5
Joint Distributions and Independence
5.1
Joint Discrete Distributions
5.2
Joint Continuous Distributions
5.3
Marginal and Conditional Distributions
5.4
Independence of Random Variables
5.5
Covariance and Correlation
5.6
Sums of Random Variables