Rodrigo Ribeiro, Ph.D.
Professor
IMPA Tech
Sala 02
Intro. Probability Theory [Summer Course]
July 1, 2020 August 1, 2020
University of Colorado Boulder
Undergraduate/Graduate
Studies axioms, combinatorial analysis, independence and conditional probability, discrete and absolutely continuous distributions, expectation and distribution of functions of random variables, laws of large numbers, central limit theorems, and simple Markov chains if time permits
1. Combinatorial Analysis
- Principle of Counting
- Permutations
- Combinations
- Number of Integer Solutions of Equations
2. Axioms of Probability
- Sample Space and Events
- Axioms of Probability
- Some Simple Propositions
- Sample Spaces Having Equally Likely Outcomes
- Probability as a Continuous Set Function
- Probability as a Measure of Belief
3. Conditional Probability and Independence
- Conditional Probabilities
- Bayes’s Formula
- Independent Events
- P(.|F) is a Probability
4. Random Variables
- Random Variables
- Discrete Random Variables
- Expected Value
- Expectation of a function of a RV
- Variance
- Bernoulli and Binomial
- Poisson RV
- Geometric RV
- Expected Value of Sums of Random Variables
- Properties of the Cumulative Distribution Function
5. Continuous Random Variables
- Continuous Random Variables
- Expectation and Variance of Continuous RV
- Uniform RV
- Normal RV
- Exponential RV
- Distribution of a Function of a RV
6. Jointly Distributed Random Variables
- Joint Distribution Functions
- Independent RV
- Sum of Independent RV: Examples
7. Properties of Expectation
- Expectation of Sums of Random Variables
- Moments of the Number of Events that Occur
- Covariance, Variance of Sums and Correlations
- Conditional Expectation
- Conditional Expectation and Prediction
- Moment Generating Function
8. Limit Theorems
- Chebyshev’s Inequality and Weak Law of Large Numbers
- Central Limit Theorem
- Law of Large Numbers