SUMMER 2018

01:640:477, Index: 01370, Section B1

Instructor: Surya Teja Gavva

Email: suryateja@math.rutgers.edu

Lectures: MTWH 8:00 am - 10:00 pm, SEC 211, Busch Campus

Course Site: Sakai- MATH PROBABILITY B1 Summer 2019

Prerequisites: Third semester calculus (Math 251 or equivalent) is an unwaivable prerequisite: a working knowledge of multiple integrals is essential.

Also, please be aware that a student can receive credit for at most one of the courses 01:640:477, 01:198:206, 01:960:381, and 14:330:321.

Textbook: Ross,

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