Course Objectives: Probability is one math topic EVERYONE has to be familiar with. In addition to beautiful mathematics, it's absolutely essential to understand many problems about nature.
The main goal of the course is to introduce basic probability and tools necessary to analyze complex probabilistic models in various fields. The focus will be on problem-solving and at the end of the course, students will have good background to go further and tackle new interesting problems.
Topics :
- Basic Counting
- Discrete Random Variables
- Continuous Distributions
- Conditional Probability and Independence
- Expectation and Variance
- Moments and Concentration Inequalities
- Central Limit Theorem(s)
Probability Puzzles
Twelvefold Way
R library for Distributions
Central limit theorem
Monty Hall problem
Probability Applets
Probability Distributions Applets
Probability and Stochastic Processes
Limits and Estimates
Sicherman Dice
Non-transitive Dice
Birthday Paradox
Monty Hall Problem
Simpson's paradox
Optimal Stopping
Class 1: Sample Space, Events, Probability
Handouts: Syllabus, Worksheet 1, Probability Puzzles
Class 2: Counting, Multiplication Principle
Permutations, Combinations, Multinomial Coefficients
Solutions to \(x_1 +x_2+\cdots +x_k=n\) (Identical balls into k boxes-Encoding trick)
Handouts: Craps, Game of Pig, Worksheet 2
Class 3: Counting review, Quiz 1
Handouts: Twelvefold way
Class 4: Conditional Probability, Multiplication rule, Total probability, Bayes Formula
Handouts: Worksheets 3, 4
Class 5: Monty Hall Problem, Simpson's Paradox, Independence, Conditional Independence, Birthday Paradox, Matching Problem, Quiz 2
Class 6: Matching Problem, Sampling Table, Random Variables, Pmf, CDF, Expectation, Variance, standard deviation, Bernoulli, Binomial random variables.
Handouts: Worksheet 5
Class 7: Random variables: Binomial, Poisson, Geometric, Negative Binomial, Hypergeometric, Quiz 3
Class 8: Gambler's Ruin, Marriage/Secretary Problem, Expected value, Indicator variables, Linearity of Expectation
Class 9: Continuous Random variables, probability density functions, Midterm Review
Class 10: MIDTERM
Class 11: Continuous rvs, pdf, CDFs, Exponential distribution, computing pdf of functions of a random variable, Uniform distribution, Normal distribution
Handouts: Worksheet 6
Class 12: Normal Distribution, Normal approximation to Binomial Random variables, Exponential distribution (Waiting time for Poisson events), Quiz 4
Class 13: Quiz 4 review, Moment Generating Functions (Moments), Joint, Marginal, Conditional distributions
Handout: Worksheet 7
Class 14: Joint, Marginal, Conditional distributions, Independence
Handout: Worksheet 8
Class 15: Sum of independent random variables, Convolutions, Quiz 5
Class 16: Expectation, Variance, Covariance, Correlation, Conditional Expectation
Handout: Worksheet 9
Class 17: Linearity of expectation, Inequalities (Union bounds, Jensens. Cauchy-Schwarz, Markov, Chebyshev)
Class 18: Chernoff bounds, Law of large numbers (Strong, Weak), Central limit theorem
Class 19: Review
Assignments:
HW0 (Not for submission)
HW1 (Due 5/30)
HW2 (Due 6/4)
HW3 (Due 6/6)
HW4 (Due 6/10)
HW5 (Due 6/17)
HW6 (Due 6/20)
HW7 (Due 6/24)
HW8 (Due 6/27)
History:
Cardano --Liber de ludo aleae
Chevalier de Méré, Pascal, Fermat -- De Méré problem, Problem of Points
Pascal --Traité du triangle arithmétique
Christian Huygens -- De ratiociniis in ludo aleae
James and Nicholas Bernoulli--
Pierre Rémond de Montmort -- Essai d'analyse sur les jeux de hasard, Matching Problem
Abraham De Moivre -- Doctrine of Chances, Normal approximation
Thomas Simpson --Doctrine of Annuities and Reversions
James Bernoulli-- Ars conjectandi, Law of large numbers
Nicholas Bernoulli, Daniel Bernoulli -- St. Petersburg problem
Poisson --
Laplace--Method of inverse probability, Théorie analytique des probabilités
Legendre --Method of least squares
Gauss -- Normal distribution
Breaking Vegas:
Beat the Wheel
Roulette Attack
Dice Dominator
Professor Blackjack
