Syllabus for Stats 2D03 (Introduction to Probability)

The following is a tentative syllabus for the course. This page will be updated regularly.
The chapters and sections refer to the 8th edition of the text book "A first course in Probability" by Sheldon Ross.
Note that the text book has all the solutions to the "self-test" problems and exercises

Week Sections in Text Suggested Homework Comments
06/09 to 07/09
Sections 1.1, 1.2, 1.3, 1.4
Chapter 1 Problems:     ##   9, 13, 15, 19, 20, 21

The first lecture this week was introductory I explained permutations and combinations, the binomial formula, Pascal's triangle, and did some examples.

10/09 to 14/09
Sections 1.5, 1.6, 2.1, 2.2, 2.3, 2.4  Chapter 1 Problems:     ##   22, 24, 26, 32
Chapter 1 Theoretical Exercises:  ##  5, 6, 8, 9, 11, 13, 16, 23
Chapter 1 Self-Test:    do all of them

Chapter 2  Problems:     ##  4, 5, 8, 12, 13, 15

Tutorials begin this week.
On Monday, and Wednesday,  I did more examples, explained the multinomial formula, the binomial tree and some consequences in various situations.  I stated Stirling's formula (not in the textbook) and finished Chapter 1.
On Thursday, I started Chapter 2 by defining sample space, sigma-algebras, events, the basic axioms of probability and simple consequences thereof, such as the exclusion-inclusion principle.
17/09 to 21/09
Sections 2.5, 2.6, 2.7, 3.1, 3.2, 3.3 Chapter 2 Theoretical Exercises:  ## 8, 9, 10, 11, 15, 16
Chapter 2  Problems:     ##  22, 27, 29, 35, 37, 38, 42, 46
Chapter 2
Self-Test:    do all of them
Chapter 3 Problems:     ##  2, 4, 12, 20, 44, 49
Chapter 3 Theoretical Exercises:  ## 1, 4, 5, 6, 7, 9

Assignment #1 was due this week on Thursday, Sept. 20th, at the beginning of the lecture period. Please hand it to me.

On Monday, I did the matching problem and a number of other problems from the textbook.
On Wednesday I did more examples from Chapter 2 and finished the Chapter with the paradoxical "12/12/12 example".
On Thursday,  I began Chapter 3 and defined the key concepts:  conditional probability, independence and Bayes' Formula which are fundamental to the subject.

24/09 to 28/09
Sections 3.4, 3.5, 4.1, 4.2, 4.3  Chapter 3  Problems:     ##  57, 67, 69, 79, 80, 81,
Chapter 3 Theoretical Exercises:  ## 12, 14, 16, 17, 21, 23, 24
Chapter 3 Self-Test:    do all of them
Chapter 4 Problems:     ## 1, 4, 5, 11, 17

On Monday and Wednesday I did more examples from Chapter 3 and finished the chapter.
On Thursday,  I began Chapter 4 and defined the fundamental notion of a random variable, its expected value (or mean) and its variance.  The Bernoulli, the Binomial and the Geometric distributions were introduced and I will define and compute the moment generating function (m.g.f.) for these distributions.
01/10 to 05/10 Sections 4.4, 4.5, 4.6, 4.7, 4.8, 4.9 Chap. 4 Problems:     ## 23, 29, 30, 32, 38, 43, 45, 46, 58, 60, 62
Chap. 4 Theoretical Exercises:  ## 1, 4, 5, 6, 7, 9, 13, 15, 16, 17

Assignment #2   was due on Thursday, October 4th,  at the beginning the lecture period.
This week,  I did more examples about computing expected values and variances of random variables.We discussed the most important discrete distributions: Bernoulli, Binomial, Hypergeometric, Geometric, Negative Binomial and Poisson. I also explained how the Poisson distribution can be viewed as a limit of the binomial distribution in a certain regime. Please read Chapter 4.

08/10 to 12/10 Sections 4.10, 5.1, 5.2, 5.3, 5.4, 5.5 Chap. 4 Problems:     ## 66, 67, 80, 84, 85
Chap. 4 Theoretical Ex.:  ## 20, 22, 24, 28, 29, 34, 35

Chap. 4 Self-Test:    do all of them
TEST #1  was held on Tuesday, Oct. 9th from 19:00 to 20:00 in T28. The test covered the material that was done in my lectures (up to last Thursday) and the material in the textbook from Chapters 1, 2, 3, and 4 up to and including 4.6
On Wednesday I explained how the hypergeometric distribution can be approximated by the binomial distribution and finished Chapter 4, by doing a number of examples and establishing all the basic facts (statistics) about discrete probability distributions that we are studying.  On Thursday, I began Chapter 5 on continuous random variables and  and introduced you to the uniform and the normal or Gaussian distribution (the mother of all distributions!)
15/10 to 19/10 Sections  5.5, 5.6, 7.7,  Chap. 5 Problems:     ## 5, 8, 10, 13, 22, 24, 29, 35, 38, 41
Chap. 5 Theoretical Ex.:  ## 1, 2, 5, 8, 11, 13, 14, 19
Chap. 5 Self-Test:    do all of them
This week I defined the  expectation and variance of continuous random variables. I explained how to normalize random variables by a shift and a scale and introduced moments and the moment generating function (m.g.f.)  I explained how the normal distribution can be viewed as a limit of a binomial distribution (Theorem of DeMoivre and Laplace).  This is a very special case of the Central Limit Theorem. The notion of the hazard function was defined and computed for the exponential and the Weibull distributions.  On Friday, I introduced the Gamma function and the Gamma distribution as waiting time for a Poisson process.  The special case of the chi-squared distribution will be interpreted as the radius squared of a standard normal in n dimensions.  Please read Chapter 5.
Click  here  and  here  and here for my lectures this week.
22/10 to 26/10 Sections  5.5, 5.6, 5.7, 6.1, 6.2
Chap. 5 Theoretical Ex.:  ## 22, 23, 24,  25, 28, 29, 31, 32
Chap. 6 Problems:  ## 1, 2, 3, 4, 5, 9, 12, 13
Chap. 6 Problems:  ## 16, 17, 22, 26, 27, 28, 29, 32, 38, 43, 48, 54
Assignment #3 was due on Monday, October 22nd (extended due date!) during the lecture period.
On Monday I did some problems from Chapter 5.  On Wednesday, I defined the Rayleigh, Cauchy, Beta and the Student-t  distributions and will do more problems from the textbook. On Thursday, I began Chapter 6 on multidimensional random variables and joint probability distribution functions and defined t and discuss in detail a discrete example involving two dice.
29/10 to 02/11 Sections  6.2, 6.3, 6.4, 6.5, 6.7
Chap. 6 Theoretical Ex.:  ## 2, 5, 6, 7, 9, 11, 14, 15, 16, 17
Chap. 6 Problems:     ## 56, 57, 59
Chap. 6 Theoretical Ex.:  ## 24, 25, 26, 35
Chap. 6 Self-Test:    do all of them
Assignment #4 was due on Thursday, November 1st during the lecture period.
On Monday,  I defined the key notion of independence of random variables, marginal distributions and prove some basic formulas, such as the expectation and variance of sums of random variables.
On Wednesday,  I  derived the convolution formula for the pdf of the sum of two random variables and proved that the sum of independent Gaussians is a Gaussian, that the sum of independent Gamma's is again a Gamma and that the sum of independent Poisson is again Poisson and did examples.
05/11 to 09/11 Sections  7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7

Chap. 7 Problems:  ## 6, 8, 9, 11, 14, 17, 25, 26, 30, 40, 43, 47
Chap. 7 Theoretical Ex.:  ## 1, 2, 4, 6, 10, 12, 16
TEST #2  was held on Tuesday, Nov. 6th from 19:00 to 20:00. in T28 (same room as Test#1) (TENTATIVELY)
The test covered the material that was done in my lectures (up to last Wednesday) and the material in the textbook from Chapters 1, 2, 3, 4, 5 and 6 up to (and including) 6.5

On Wednesday, I explained the change of variables formula for multivariate distributions and finish Chapter 6.  On Thursday started Chapter 7. As you might notice, I have already covered some of the material from Chapter 7 earlier, such as the moment generating function and covariance.
12/11 to 16/11
Sections 7.8, 7.9, 8.1, 8.2, 8.3, 8.4, 8.5

Chap. 7 Problems:  ## 50, 53, 56, 62, 64, 70, 71, 72, 75, 76
Chap. 7 Theoretical Ex.:  ## 20, 24, 26, 27, 38, 39
Chap. 7 Self-Test:    do all of them
Chap. 8 Problems:  ## 2, 4, 6, 7, 9, 11, 13, 14, 19, 20, 21, 23.
On Monday and Wednesday I discussed basic estimators: sample mean and sample variance and how to build confidence intervals using the student t, which I derived as a ratio of Z and a (scaled) chi squared distribution.
On Thursday I began Chapter 8 and
proved Markov's, Chebyshev's and the Law of Large Numbers
inequalities.

19/11 to 23/11

Sections 8.6
Section 9.2

Chap. 8 Theoretical Exercises: do all of them
Chap. 8 Self-Test:  do all of them

Assignment #5  was due this week on Monday. Please hand it to me at the beginning of the lecture period.
On Monday, I proved
the Central Limit Theorem.
On Wednesday I did some more inequalities: Jensen's  and Chernoff's.
On Friday I explained the famous Black-Scholes formula.
26/11 to 30/11  REVIEW

This week will be spent tidying up some loose ends and reviewing the material I taught.
GOOD-BYE