Further Topics in Probability, Spring 2022

MATH30006, MATHM0018

This unit will run face-to-face.


 
Márton Balázs
Email:m.balazs@our_city.ac.countrycode
Office:1.44 Fry
Drop in Sessions: Tuesdays 16:30 - 17:30 in 1.44 Fry, and Wednesdays 18:00 - 19:00 on Zoom (see link on Blackboard)
Classes and support sessions: See in your timetables (and Blackboard about cancellations, etc)
Q&A Session: Wednesday 18th May, 3:00pm, G.10. 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 of an approved type (permissible for A-Level examinations) are permitted in the examination.
     
  • This unit will happen face-to-face, and you are expected to follow lectures in the classroom. Nevertheless, I will link YouTube videos from previous years on the material, see the column "Past videos" in the table. I hope YouTube works for everyone interested, let me know otherwise.
     
  • I will assume familiarity with the following concepts from calculus and analysis, but of course I'm happy to discuss these outside class:
    • basic properties, derivatives and integrals of elementary functions like polynomials, exponential, logarithm, trigonometric and some hyperbolic functions
    • the definition of limit, liminf and limsup of real-valued sequences
    • that monotone sequences have limits
    • the definition of a Cauchy sequence and that these are convergent in ℝ
    • that 1/np is summable for p>1 but not for p≤1, and the analogous statement for integrals
    • that exponentials converge/diverge faster than any polynomial
    • the triangle inequality for absolute value
    • the way to find maxima and minima of smooth real to real functions
    • the Heine–Cantor theorem on uniform continuity of continuous real-valued functions on a closed and bounded real interval
    • complex numbers, and exponentials thereof
    • Cauchy's Residue Theorem. This will be stated in class and in the notes together with examples of how to use it, so don't worry about it too much.

     
  • 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. Our marking deadline is 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 exams at levels H/6 and M/7 will have cca. 80% overlap. Past exams, one with solutions, are to be found on the Blackboard page Resources for students → Examinations.
     
  • The standard normal distribution in pdf. A similar table will be available on the exam.
     
  • Revision notes in pdf, written by Aaron Smith, a student in this unit in 2015. The notes on Stirling's formula and the DeMoivre-Laplace CLT, and a product-sum lemma are additional to this. 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: 14/02/2022.)
     
  • 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.)
     
  • I came across this illustration of Jensen's inequality: square-root of the average ≠ average of the square-root.
     
  • I was thinking about going over the ingenious proof of the SLLN by N. Etemadi, or this other one using ergodic ideas by C.W. Chin, but decided to stay with the classical argument instead. You are welcome to check out their ways of doing it!
     
  • Here is an alternative way of getting to the Continuity theorem by Christian Döbler, in case you are interested.

Below is a detailed schedule. Topics of future events are plans, and can change. Topics of past events serve as log. The green version of pdf's use less paper to print. Previous years' videos are are added for convenience, and the recommended literature can be found under the unit description links above. Both of these are advisory, examinable is what is featured in lectures.

Homeworks are/will also be posted here: just click those with a link below. They are due every second Thursday (see below) at 12:00pm in Blackboard. Homework solutions will appear on Blackboard, please email me if you have problem accessing these.


 
Time Topics Past videos Homework due:
Mon 24 Jan Intorduction; Basic discrete distributions L1, L2
Tue 25 Jan Convolution (discrete cases)
Basic continuous distributions 		L3, L4, L5, L6
L7 		--
Fri 28 Jan Ex. class (convolution)
Mon 31 Jan Normal distribution
Convolution (Uniform, Gaussian)
 L8
L9, L10
Tue 1 Feb Convolution (Cauchy) L11, L12, L13 by Thu 3 Feb, noon:
HW1 (sol. on Bb.)
Fri 4 Feb Convolution (Cauchy)
Mon 7 Feb Gamma, Chi square distributions L14, L15, L16
Tue 8 Feb Poisson process
Generating functions (properties) 		L17
L18 		--
Fri 11 Feb Ex. class (generating function examples) L19, L20, L21, L23
Mon 14 Feb Generating functions (random no. of summands) L24, L25
Tue 15 Feb Generating functions (Galton-Watson process) L26, L27 by Thu 17 Feb, noon:
HW2 (sol. on Bb.)
Fri 18 Feb Critical G-W process; total population L28, L29, L30
Mon 21 Feb Generating functions (random walk: level 1 hitting time) L31, L32
Tue 22 Feb Generating functions (random walk: recurrence and transience) L33 --
Fri 25 Feb Ex. class (fun with generating functions: triangle, Neg.Binomial) L22
Mon 28 Feb Generating functions (weak convergence) L34
Tue 1 Mar Generating functions (weak convergence, Law of Rare Events) L35, L36 by Thu 3 Mar, noon:
HW3 (sol. on Bb.)
Fri 4 Mar Generating functions (Law of Rare Events) L37
Mon 7 Mar Weak convergence
Weak Law of Large Numbers
Stirling's formula 		L38, L39
Tue 8 Mar Stirling's formula L40, L41 --
Fri 11 Mar Ex. class (RW probabilities (sol. on Bb))
Mon 14 Mar DeMoivre Laplace CLT L42, L43
Tue 15 Mar DeMoivre Laplace CLT L44, L45 by Thu 17 Mar, noon:
HW4 (sol. on Bb.)
Fri 18 Mar DeMoivre Laplace CLT
(Review from Martingales: Measure Theory, product-sum lemma, probabilistic tools, convergence modes) 		L46
(M1, M2, L47, M3, M4, M5, M6,
F1, F2, F3, F4, F5, F6, F7, F8,
F9, F10, F11, F12, E2, E3, E4, E5)
Mon 21 Mar SLLN (Kolmogorov's ineq.; Kolmogorov-Khinchin) F13, F14, F15
Tue 22 Mar SLLN (Toeplitz' lemma; Kronecker; Kolmogorov's Thm) F16, F17
F18 		--
Fri 25 Mar Ex. class (SLLN)
Mon 28 Mar SLLN (Kolmogorov's Thm, final proof, remarks) F19, F20, F21, F22
Tue 29 Mar Characteristic Functions F23, F24, F25, F26 --
Fri 1 Apr Inversion formula F27, F28
Easter vacation
Mon 25 Apr Inversion formula, weak convergence F29, F30 by Mon 25 Apr, noon:
HW5 (sol. on Bb.)
Tue 26 Apr Weak Convergence (Prokhorov's Thm) F31, F32
Fri 29 Apr Ex. class (Characteristic functions (sol. on Bb))
Tue(!) 3 May Continuity Lemma F33, F34, F35, F36
Tue 3 May WLLN, CLT F37, F38, F39 by Thu 5 May, noon:
HW6 (sol. on Bb.)
Fri 6 May Comments on the CLT, Berry-Esseen, Local CLT, Lindeberg Thm F40, F41, F42

 
 

 

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