Further Topics in Probability, Spring 2016

Márton Balázs
Tel:+44 (0)117 928-7991
Office:Maths 3.7
Drop in Sessions: Tuesdays 10:30 - 11:30 on teaching weeks, Maths 3.7
except for the 8th of March, which is cancelled.
Q&A Session: Friday 20th May, 10:00am, SM2.

  • 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 of an approved type (non-programmable, no text facility) are 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. 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.
  • 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:
Mon 25 Jan Review of the unit, basic discrete distributions --
Tue 26 Jan Convolution (discrete cases) --
Thu 28 Jan Ex. Class (Neg.bin., Poi., Uniform convolutions) --
Mon 1 Feb Normal distribution, Convolution (Gaussian) --
Tue 2 Feb Convolution (Cauchy) --
Thu 4 Feb Convolution (Cauchy, Exponential) HW1 (sol. on Bb.)
Monday's class rescheduled to Tuesday
Tue 9 Feb Poisson process, Chi square --
Tue 9 Feb Generating functions (properties; examples; of convolutions) --
Thu 11 Feb Ex. Class (Moran's example, neg.bin.+combi. generating fcts) --
Mon 15 Feb Generating functions (random no. of summands) --
Tue 16 Feb Generating functions (Galton-Watson process) --
Thu 18 Feb Generating functions (Critical G-W process; random walk: level 1 hitting time) HW2 (sol. on Bb.)
Mon 22 Feb Generating functions (random walk) --
Tue 23 Feb Generating functions (Weak convergence, Poisson approximation) --
Thu 25 Feb Ex. Class (Generating functions: Catalan example, two sixes problem) --
Mon 29 Feb Weak convergence examples, Weak Law of Large Numbers --
Mon 29 Feb Weak Law of Large Numbers, Stirling's formula --
Tue 1 Mar Stirling's formula, DeMoivre Laplace CLT --
Thu 3 Mar Ex. Class (Generating functions: RW probabilities) HW3 (sol. on Bb.)
I'm away
Mon 14 Mar DeMoivre Laplace CLT --
Mon 14 Mar Measure Theory (basic notions, probability) --
Tue 15 Mar Measure Theory, Probabilistic tools --
Tue 15 Mar Probabilistic tools --
Thu 17 Mar Probabilistic inequalities; types of convergences HW4 (sol. on Bb.)
Easter vacation
Mon 11 Apr Convergences --
Tue 12 Apr Strong Law of Large Numbers (Kolmogorov's ineq., Kolmogorov-Khinchin) --
Thu 14 Apr Ex. Class (Gen. fcts; convergence types) --
Mon 18 Apr Toeplitz, Kronecker; Kolmogorov's Thm, SLLN --
Tue 19 Apr Characteristic Functions --
Thu 21 Apr Characteristic Functions (Inversion formula) HW5 (sol. on Bb.)
Mon 25 Apr Inversion formula and its consequences --
Tue 26 Apr Weak Convergence, CLT (Prokhorov's Thm) --
Thu 28 Apr Ex. Class (Characteristic functions) --
Mon 2 May Bank holiday
Tue 3 May Weak Convergence, CLT (Continuity Lemma) --
Wed 4 May Weak Convergence, WLLN, CLT --
Thu 5 May Weak Convergence, CLT (Comments on the CLT) HW6 (sol. on Bb.)



If you have any questions, please contact me (see on top). Below is my schedule.



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