1 Introduction To Probability
1.1 Introduction
1.2 Notation and Terminology
1.3 Definitions and Axioms of Probability
1.3.1 Axioms of probability
1.3.2 Practice problems
1.4 Independent Events, Conditional Probability and Bayes’ Theorem
1.4.1 Independent Events
1.4.2 Conditional Probabilities
1.4.3 Bayes theorem
1.4.4 Contingency Tables
1.4.5 Practice problems
1.5 Counting Techniques
1.5.1 Multiplication Principle
1.5.2 Permutations
1.5.3 Combinations
1.5.4 Practice problems
2 Random Variables and Probability Distributions
2.1 Random Variables
2.1.1 Summation methods you will need
2.1.2 Practice problems
2.2 Discrete Random Variables
2.2.1 Practice problems
2.2.2 Practice problems
2.3 Continuous Random Variables
2.3.1 Practice problems
2.4 Mathematical Expectations and Generating Functions of Random Variables
2.4.1 Expectations and Moments
2.4.2 Probability Generating Functions (PGFs)
2.4.3 Moment Generating Functions (MGFs)
2.4.4 Characteristic Functions
2.4.5 Practice problems
2.5 Some Important Discrete Probability Mass Function
2.5.1 The Bernoulli Distribution
2.5.2 Discrete Uniform
2.5.3 The Binomial Distribution
2.5.4 The Hypergeometric Distribution
2.5.5 The Geometric Distribution
2.5.6 The Negative Binomial Distribution
2.5.7 The Poisson Distribution
2.5.8 The Zeta (or Zipf) Distribution
2.5.9 Practice problems
2.6 Some Important Continuous Probability Density Functions
2.6.1 The Uniform Distribution
2.6.2 Exponential Distribution
2.6.3 Gamma Distribution
2.6.4 Beta Distribution
2.6.5 Normal Distribution
2.6.6 Cauchy Distribution
2.6.7 The Weibull Distribution
2.6.8 Rayleigh Distribution
2.6.9 The Pareto Distribution
2.6.10 Practice problems
2.7 Cumulative Distribution Function (CDF)
2.7.1 Practice problems
2.8 Markov’s and Chebyshev’s Inequality
2.8.1 Practice problems
3 Jointly Distributed Random Variables
3.1 Introduction
3.2 Joint Distributions
3.2.1 Joint Cumulative Distribution Function
3.2.2 Expected Values
3.2.3 Practice problems
3.3 Conditional Probability Functions and Independence of Random Variables
3.3.1 Special Joint Distribution Function (Bivariate Normal)
3.3.2 Practice problems
3.4 Joint Moment Generating Function