Further Topics in Probability, Spring 2017

Márton Balázs
Tel:+44 (0)117 928-7991
Office:Maths 3.7
Drop in Sessions: Tuesdays 17:00 - 18:00 on teaching weeks, Maths 3.7
Q&A Session: Friday 19th May, 14:00, SM2. 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. On the exam, you can collect 100 exam marks. Your final score will be computed using the formula [homework marks]*0.2 + [exam marks]*0.8. This score will be compared to the standard 0-100 scale. (Yes, you have a gift value of 0.4.)
  • 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: 03/02/2017.)
  • 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.)

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 24 Jan Review of the unit, basic discrete distributions --
Wed 25 Jan Convolution (discrete cases) --
Thu 26 Jan Ex. class (Neg.Bin convolution, continuous r.v., Normal distribution) --
Tue 31 Jan Convolution (Gaussian) --
Wed 1 Feb Convolution (Cauchy) --
Thu 2 Feb Convolution (Cauchy, Exponential) HW1 (sol. on Bb.)
Tue 7 Feb Gamma, Chi square distributions --
Wed 8 Feb Poisson process. Generating functions (properties) --
Thu 9 Feb Ex. class (convolutions) --
Tue 14 Feb Generating functions (random no. of summands) --
Wed 15 Feb Generating functions (Galton-Watson process) --
Thu 16 Feb Generating functions (Critical G-W process; random walk: level 1 hitting time) HW2 (sol. on Bb.)
Tue 21 Feb Generating functions (random walk) --
Wed 22 Feb Generating functions (Weak convergence, Poisson approximation) --
Thu 23 Feb Ex. class (fun with generating functions) --
Tue 28 Feb Weak convergence examples, Weak Law of Large Numbers --
Wed 1 Mar Weak Law of Large Numbers, Stirling's formula --
Thu 2 Mar Stirling's formula, DeMoivre Laplace CLT HW3 (sol. on Bb.)
Tue 7 Mar DeMoivre Laplace CLT --
Wed 8 Mar Measure Theory (basic notions, probability) --
Thu 9 Mar Ex. class (Generating functions: RW probabilities) --
Tue 14 Mar Measure Theory --
Wed 15 Mar Probabilistic tools --
Thu 16 Mar Probabilistic inequalities HW4 (sol. on Bb.)
Tue 21 Mar Convergences --
Wed 22 Mar Strong Law of Large Numbers (Kolmogorov's ineq., Kolmogorov-Khinchin) --
Thu 23 Mar Toeplitz, Kronecker; Kolmogorov's Thm, SLLN --
Tue 28 Mar Ex. class (convergence types) --
Wed 29 Mar SLLN; characteristic Functions --
Thu 30 Mar Characteristic Functions (Inversion formula) --
Easter vacation
Tue 25 Apr Inversion formula and its consequences HW5 (sol. on Bb.)
Wed 26 Apr Weak Convergence, CLT (Prokhorov's Thm) --
Thu 27 Apr Ex. class (Characteristic functions) --
Tue 2 May Weak Convergence, CLT (Continuity Lemma) --
Wed 3 May Weak Convergence, WLLN, CLT --
Thu 4 May Comments on the CLT (Berry-Esséen, Local CLT, Lindeberg Thm) HW6 (sol. on Bb.)



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


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