Further Topics in Probability, Spring 2018

Márton Balázs
Tel:+44 (0)117 928-7991
Office:Maths 3.7
Drop in Sessions: Tuesdays 13:00 - 14:00 on teaching weeks, Maths 3.7
Q&A Session: Friday 25th May, 14:00, SM4. Please prepare with questions.

  • The unit description for level H/6, including assessment methods, texts, syllabus.
  • The unit description for level M/7, including assessment methods, texts, syllabus.
  • Calculators are not allowed in the examination.
  • Remark on the assessments: there will be no distinction between levels H/6 and M/7 regarding homeworks. Six homework sets will be assigned, see the schedule below. My marking deadline is the class one week after the respective due dates. From each of these, you can collect 17 homework marks. Your final mark will be weighted as 20% -- 80% homework mark -- exam mark.
  • A few more remarks on the exam: for both levels, final examination will be 2½-hour long, will consist of four questions each of which will be used for assessment. The difference between levels H/6 and M/7 will only be the required depth of reproducing proofs in the exam. On H/6 I can ask for at most a couple of words about how a proof goes. On M/7, I can ask for proofs in details.
  • The standard normal distribution in pdf. You will be given a similar table on the exam.
  • Revision notes in pdf, written by Aaron Smith, a student in this unit in 2015. Please notice that these notes are by no means meant to fully cover our material, nor will all parts of them be assessed. (Last edited: 25/05/2018.)
  • Extended slides/notes of elementary probability. Some of it can be used as reference and refreshment for parts of the Probability 1 and Advanced Probability 2 units. Other portions we will cover in class, and some we will not touch. Below you'll see links to relevant parts of this material. Please notice that these links are by no means meant to fully cover our material, nor will all parts of all of them be assessed. They rather serve as background reading. (They are actually part of lecture notes for a rather strong first probability unit I used to teach before.)
  • A mock exam with solutions is now available on Blackboard. Disclaimer: no two exams are the same, this is just to show you some possible style of how our exam could look like. The topics and difficulty of the questions might vary. Concentrating on past and mock exams is in general NOT a good way of preparation.

Below is a detailed schedule. Topics of future events are plans, and can change. Topics of past events serve as log.
Homeworks are/will also be posted here: just click those with a link below. They are due in class, or in the blue locker with my name on the ground floor of the Main Maths Building.
Homework solutions will appear on Blackboard, please email me if you have problem accessing these.

I was thinking about going over the ingenious proof of the SLLN by N. Etemadi, but decided to stay with the classical way instead. You are welcome to check out his way of doing it!

Came across this illustration of Jensen's inequality: square-root of the average ≠ average of the square-root.

Day Topics Homework due:
Tue 23 Jan Review of the unit, basic discrete distributions --
Thu 25 Jan Basic discrete distributions; convolution (discrete cases) --
Fri 26 Jan Ex. class (Neg.Bin convolution, continuous r.v., Normal distribution) --
Tue 30 Jan Convolution (Gaussian) --
Thu 1 Feb Convolution (Cauchy) --
Fri 2 Feb Convolution (Cauchy) HW1 (sol. on Bb.)
Tue 6 Feb Gamma, Chi square distributions --
Thu 8 Feb Poisson process. Generating functions (properties) --
Fri 9 Feb Ex. class (convolutions) --
Tue 13 Feb Generating functions (random no. of summands) --
Thu 15 Feb Generating functions (Galton-Watson process) --
Fri 16 Feb Generating functions (Critical G-W process; random walk) HW2 (sol. on Bb.)
Tue 20 Feb Generating functions (random walk: level 1 hitting time) --
Thu 22 Feb Generating functions (weak convergence) --
Fri 23 Feb Ex. class (fun with generating functions) --
Tue 27 Feb Weak convergence examples, Weak Law of Large Numbers --
Thu 1 Mar Weak Law of Large Numbers, Stirling's formula -- Cancelled --
Fri 2 Mar Stirling's formula, DeMoivre Laplace CLT -- Cancelled HW3 -- Due: 5th (sol. on Bb.)
Tue 6 Mar Weak Law of Large Numbers, Stirling's formula --
Thu 8 Mar Weierstrass' Approximation Thm, DeMoivre Laplace CLT --
Thu 8 Mar DeMoivre Laplace CLT --
Fri 9 Mar Ex. class (Generating functions: RW probabilities) --
Tue 13 Mar Measure Theory (basic notions, probability) --
Tue 13 Mar Measure Theory --
Thu 15 Mar Probabilistic tools --
Fri 16 Mar Probabilistic inequalities HW4 (sol. on Bb.)
Tue 20 Mar Convergences --
Thu 22 Mar Strong Law of Large Numbers (Kolmogorov's ineq., Kolmogorov-Khinchin) --
Fri 23 Mar Ex. class (convergence types) --
Easter vacation
Tue 17 Apr Toeplitz, Kronecker; Kolmogorov's Thm, SLLN --
Thu 19 Apr SLLN; characteristic Functions --
Fri 20 Apr Characteristic Functions HW5 (sol. on Bb.)
Tue 24 Apr Inversion formula --
Thu 26 Apr Inversion formula, weak convergence (Prokhorov's Thm) --
Fri 27 Apr Ex. class (Characteristic functions) --
Tue 1 May Weak Convergence (Prokhorov's Thm) --
Thu 3 May Weak Convergence (Continuity Lemma) --
Fri 4 May WLLN, CLT, comments HW6 (sol. on Bb.)



If you have any questions, please contact me (see on top). Click here to see my schedule.


To the teaching page