Math 461 Probability Theory

Renming Song Fall 2021

• Syllabus
  

• Homework
  

• Sample Tests
  

Sample Test 1

Sample Test 1

Sample Test 2

Sample Test 2

Sample Final

Sample Final

• Solution to Homework Assignments
  

Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7

Homework 8

Homework 9

Homework 10

Homework 11

• Lecture Slides
  

08/23

08/25

08/27

08/30

09/01

09/03

09/08

09/10

09/13

09/15

09/17

09/20

09/22

09/24

09/27

09/29

10/01

10/04

10/06

10/11

10/13

10/15

10/18

10/20

10/22

10/25

10/27

10/29

11/01

11/03

11/05

11/08

11/10

11/15

11/17

11/19

11/29

12/01

12/03

12/06

12/08

• Solution to Tests
  

Test 1

Test 2

• Class Diary

• Monday, 08/23: Basic principle of counting (multiplication rule), permutations, combinations.
  Read: Sections 1.2, 1.3 and 1.4
• Wedensday, 08/25: combinations, binomial theorem, multinomial coefficients, multinormial theorem, number of integer solutions of equations
  Read: Sections 1.4, 1.5 and 1.6.
• Friday, 08/27: sample space, events, axioms of probability, simple properties of probability measures
  Read: Sections 2.2, 2.3, 2.4
• Monday, 08/30: sample spaces having equally likely outcomes.
  Read: Section 2.5.
• Wedensday, 09/01: sample spaces having equally likely outcomes, conditional probability.
  Read: Sections 2.5 and 3.2
• Friday, 09/03: Conditional probability, Bayes rule
  Read: Sections 3.2 and 3.3
• Wedensday, 09/08: Independent events, independent trails.
  Read: Section 3.4
• Friday, 09/10: problem of points, gambler's ruin, random variables.
  Read: Sections 3.4 and 4.1
• Monday, 09/13: random variables, distribution functions, discrete randim variables,probability mass functions, properties of distribution functions
  Read: Section 4.1, 4.2, 4.10
• Wedensday, 09/15: Expectation, expectation of a function of a discrete random variable, moments, variance.
  Read: Sections 4.3, 4.4, 4.5
• Friday, 09/17: Bernoulli random variables, binomial random variables, Poisson random variables
  Read: Sections 4.6 and 4.7
• Monday, 09/20: Poisson random variables, geometric random vraibles
  Read: Sections 4.7 and 4.8.
• Wedensday, 09/22: negative binomial random variables, expectation of sum of random variables.
  Read: Sections 4.8 and 4.9
• Friday, 09/24: expectation of sum of random variables, Continuous random variables, densities, absolutely continuous random variables.
  Read: Sections 4.9 and 5.1
• Monday, 09/27: absolutely continuous random variables, expectations of absolutely continuous random variables, Eexpectation of a function of an absolutely continuous random variable.
  Read: Sections 5.1 and 5.2
• Wedensday, 09/29: variance of an absolutely continuous random variable, uniform random variables, standard normal density
  Read: Sections 5.2, 5.3 and 5.4
• Friday, 10/01: normal random variables
  Read: Section 5.4
• Monday, 10/04: DeMoivre-Laplace central limit theorem, application to polling.
  Read: Section 5.4
• Wedensday, 10/06: review,
  Read:
• Friday, 10/08: Test 1. 95-99: 2, 90-94: 1, 85-89: 2, 80-84: 2, 75-79: 3, 70-74: 0, 60-69: 2, -59: 2
• Monday, 10/11: Exponential random variables, memoryless property, Gamma function
  Read: Sections 5.5 and 5.6
• Wedensday, 10/13: Gamma random variables, Beta random variables, the distribution of a function of a random variable
  Read: Sections 5.6 and 5.7
• Friday, 10/15: distribution of a function of a random variable, joint distributions, marginal distributions, joint mass functions, marginal mass functions, joint densities, marginal densities
  Read: Sections 5.7 and 6.1
• Monday, 10/18: Joint distribution of n random variables, joint mass function of n random vaiables, joint density of n random variables, multinomial distribution, independent random variables.
  Read: Sections 6.1 and 6.2
• Wedensday, 10/20: independent random variables, Sums of independent random variables.
  Read: Sections 6.2 and 6.3
• Friday, 10/22: Sums of independent random variables.
  Read: Section 6.3
• Monday, 10/25: Conditional distributions.
  Read: Sections 6.4 and 6.5
• Wedensday, 10/27: Order statistics, expectation of a function of random variables.
  Read: Sections 6.6 and 7.2
• Friday, 10/29: Covariance, correlation coefficients, properties of covariances.
  Read: Section 7.4
• Monday, 11/01: Variance of sums
  Read: Section 7.4
• Wedensday, 11/03: Conditional expectation
  Read: Section 7.5
• Friday, 11/05: Conditional expectation, moment generating functions
  Read: Sections 7.5 and 7.7
• Monday, 11/08: moment generating functions
  Read: Section 7.7
• Wedensday, 11/10: Review
• Friday, 11/12: Test 2: 95-99: 1, 90-94: 1, 85-89: 1, 80-84: 1, 75-79: 2, 70-74: 1, 60-69: 1, 50-59: 1, 40-49: 1, -39: 1
• Monday, 11/15: Markov inequality, Chebyshev inequality, weak law of large numbers
  Read: Section 8.2
• Wedensday, 11/17: Central limit theorem
  Read: Section 8.3
• Friday, 11/19: strong law of large numbers
  Read: Section 8.4
• Monday, 11/29: Review,
  Read:
• Wedensday, 12/01: Review
• Friday, 12/03: Review
• Monday, 12/06: Review
• Wedensday, 12/08: Review