Lectures: MWF 1:00 - 1:50pm in MSTB 124

Discussion: TuTh 8:00 - 8:50am in ICS 180

Course description: Joint distributions, sums of independent random variables, conditional expectation, covariance, moment generating functions, multivariable normal distribution, limit theorems, entropy. Chapters 6 - 9 of the book will be covered.

Prerequisites: MATH 130A or MATH 131A or STATS 120A.

Textbook: S. Ross, First Course in Probability, 9th edition. ISBN: 9780321794772

The course grade will be determined as follows:

• Homework: 10%. One homework with the lowest score will be dropped. Solutions will be collected every Thursday. Late homework will not be accepted. You are welcome and encouraged to form study groups and discuss homework with other students, but you must write your solutions individually.
• Quizzes: 15%, every Thursday. One quiz with the lowest score will be dropped.
• Midterm Exam 1: 20%, Wednesday, January 31, in class.
• Midterm Exam 2: 20%, Friday, February 23, in class.
• Final Exam: 35%, Wednesday, March 21, 1:30 - 3:30pm.

There will be no make-up for quizzes or exams for any reason. A missed midterm exam counts as zero points, with the following exception. If you miss a midterm exam due to a documented medical or family emergency, the exam's weight will be added to the weight of the final exam.

• Monday, January 8

  Review of Sections 2-5.

• Wednesday, January 10

  Joint and marginal distributions (6.1).
  Homework 1 (due January 18) from Chapter 6: 1(a), 2(a), 6, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 27.

• Friday, January 12

  Independent random variables (6.2).

• Wednesday, January 17

  Independent random variables cont. (6.2). Buffon's needle.
  Homework 2 (due January 25) from Chapter 6: 21, 22, 23, 24, 26, 29, 30, 31, 32.

• Friday, January 19

  Sums of independent random variables (6.3). Convolution. Sums of independent normal random variables (6.3.3).

• Monday, January 22

  Applications to sample mean. Sums of independent Bernoulli, Binomial, Poisson and exponential random variables (6.3.4, 6.3.2).

• Wednesday, January 24

  Gamma and chi-square distributions (6.3.2).
  Homework 3 (do not turn in): practice for Midterm Exam 1. Review past homework problems and do the following self-test problems at the end of Chapter 6: 1, 2, 3, 5, 6, 7, 8, 12, 13, 16.

• Friday, January 26

  Conditional distributions and conditional expectations: discrete case (6.4). The Exchange Paradox.

• Monday, January 29

  Conditional distributions and conditional expectations: continuous case (6.5). A version of Bayes Formula for continuous distributions.

• Wednesday, January 31

  Midterm Exam 1.
  Solutions
  Homework 4 (due February 8) from Chapter 6: 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 55, 56(a), 58; theoretical exercises 19, 24.

• Friday, February 2

  Transformations of joint distributions (6.7). Cauchy distribution as a ratio of two normals.

• Monday, February 5

  Properties of expectation (7.1). Expectation of sums of random variables (7.2-7.3). Bernoulli's example about deaths in town.

• Wednesday, February 7

  Further applications of linearity of expectation: sum of the angles in a triangle, cutting a graph, balancing vectors, and finding independent sets in a graph.

• Friday, February 9

  Expectation of product of independent random variables. Covariance and correlation (7.4).
  Homework 5 (due February 15) from Chapter 7: 3, 4, 5, 9 (you may leave sums and products in the answer), 13, 15, 21, 23, 27, 30, 38.

• Monday, February 12

  Covariance and correlation (7.4): examples, pitfalls of correlation (relation with independence, correlation does not imply causation, correlation can be caused by confounding variables).

• Wednesday, February 14

  Properties of covariance. Variance of a sum of random variables (7.4).
  Homework 6 (due February 22) from Chapter 7: 26, 31, 32, 33, 34, 36, 37, 39, 41, 42 (first part), 45, 46, 63(a).

• Monday, February 19

  Sample mean and sample variance (Example 4a).

• Wednesday, February 21

  Conditional expectation (7.5).
  Homework 7 (due March 1) from Chapter 7: 48, 50, 51, 55, 62 (in part b, you may use the formula for expectation from Exercise 5 in Chapter 4), 64(a), 68(a,b), 69.

• Friday, February 23

  Midterm Exam 2.
  Solutions

• Monday, February 26

  Moment generating functions (7.7).

• Wednesday, February 28

  Moment generating functions (cont.)
  Homework 8 (due March 8) from Chapter 7: 61, 65, 66 (study Example 5c first), 70, 71, 72 (in part c, you may use the formula for expectation from Theoretical Exercise 5 in Chapter 4), 75 (recognize the distributions of X and Y by name first), 76 (see definition of joint MGF in Section 7.7.1), 77, Theoretical exercises 46, 50.

• Friday, March 2

  Joint moment generating functions. Multivariate normal distribution.

• Monday, March 5

  Random vectors. Covariance matrices.

• Wednesday, March 7

  Density of the multivariate normal distribution.

• Friday, March 9

  Characteristic functions. Fourier transform. Central limit theorem (8.3).
  Homework 9 (due March 15) from Chapter 6: 12, 13, 28, 48, 57; from Chapter 7: 7, 19, 40, 47, 56, 79(a,b); from Chapter 8: 5, 7, 10, 13.

• Monday, March 12

  Application of the central limit theorem: Brownian motion.

• Wednesday, March 14

  The strong law of large numbers (8.4). Borel-Cantelli lemma. Application: infinite monkey theorem.

• Friday, March 16

  Further applications of the law of large numbers: Monte-Carlo method, Weierstrass approximation theorem. Frequentist and Bayesian approaches to probability.