ST2004 - Applied Probability I 


2018/2019


Module Code: ST2004

Module Title: Applied Probability I (module descriptor)

Credit Value: 5

Cohorts: MSISS, CSL(JS)

Week range: 03-08, 10-14 

Semester: S1 (Michaelmas) 

Total hours: 33 (28 classes + 5 labs)


Textbook: Tijms, "Understanding Probability" (pages indicated below refer to 3rd edition).

Additional material will be provided when needed.


Schedule:

Monday, 9 am - 10 am, Lab ICT 1 (weeks 3-8), LB107 (weeks 10-14)

Tuesday, 5 pm - 6 pm, LB08

Wednesday, 3 pm - 4 pm, LB04


Group assignment (password protected)

  • instructions (pdf), deadline: Monday 3/12/18


Material (password protected):

  • 11/09/18: slides (pdf)
  • 12/09/18: slides (pdf)
  • 18/09/18: slides (pdf)
  • 09/10/18: slides (pdf), excel worksheet (xlsx)
  • 12/11/18: list of examinable material (pdf)
  • 12/11/18: mock exam (pdf)
  • 12/11/18: doc-cam notes (pdf)


Labs:

  • 17/09/18: lab 1 - instructions (pdf) and template (xlsx)
  • 24/09/18: lab 2 - instructions (pdf) and template (xlsx)
  • 01/10/18: lab 3 - instructions (pdf) and template (xlsx)
  • 08/10/18: lab 4 - instructions (pdf) and template (xlsx)
  • 15/10/18: lab 5 - instructions (pdf) and template (xlsx)


Syllabus (lecture by lecture):

  • 11/09/18: Monte Carlo approach, Empirical Law of Large Numbers, Birthday problem (via simulation).

Textbook: Section 2 (pp. 18-25), Section 3 (pp. 75-80, 3.1.4 excluded).

  • 12/09/18: True and pseudo random number generation, Multiplicative congruential method, Random generation from an interval, Random generation from integers.

Textbook: Section 2.8 (pp. 52-59), Section 2.9.4 (pp. 62-63).

  • 17/09/18: Lab 1: generating random passwords.
  • 18/09/18: Generation of random permutations, 1970 draft lottery problem.

Textbook: Section 2.8.1 (pp. 55-56), Section 3.7 (pp. 98-102).

  • 19/09/18: Frequentist probability, Axiomatic foundations of probability.

Textbook: Chapter 7, Section 7.1 (pp. 229-232).

  • 24/09/18: Lab 2: does skill win the league? 
  • 25/09/18: Definition of probability functions on a finite or countably infinite sample space, Equally likely events, Examples.

Textbook: Section 7.1.1 (pp. 232-235). 

  • Assignment: Problems 7.1-7.5 and 7.14 (if you have doubts, ask at the tutorial of the 03/10/18).
  • 26/09/18: Probability on an uncountable sample space (dartboard example)Derivation of basic rules of probability from axioms (with proofs).

Textbook: Example 7.3 (pag. 234), Section 7.2 (pp. 239-241, 7.2.1 excluded), Section 7.3 (pp. 243-244). 

Assignment: Problems 7.27 and 7.28 (if you have doubts, ask at the tutorial of the 03/10/18).

  • 01/10/18: Lab 3: Normal distribution and systems' lifetime.
  • 02/10/18: Independent event, Conditional probability, Bayes rule
Textbook: Chapter 6 (pp. 220-222), Chapter 8 (pp. 256-258, pp. 260-261, pp. 264-265).

Assignment: Problems 8.1, 8.2 and 8.3.
  • 03/10/18: Tutorial: factorial number and binomial coefficient, solutions to Problems 7.3, 7.5 and 7.27.
Textbook: Appendix (pp. 532-535).
  • 08/10/18: Lab 4: summaries of variation.
  • 09/10/18: Law of conditional probability, Bayes theorem, Monty Hall problem
Textbook: Chapter 6 (pp. 212-217, 6.1.3 excluded), Chapter 8 (pp. 266-267).

Assignment: Problems 8.17, 8.31, 8.32.
  • 10/10/18: Application of Bayes Theorem: Problems 8.19 and 8.30, Random variable (RV), Discrete RV, Range of a discrete RV, Probability mass function (pmf) of a discrete RV.
Textbook: Chapter 9, pp. 283-286.

Assignment: study the pmf of the max and the min of the toss of two dice.
  • 15/10/18: Lab 5: Simple Queues.
  • 16/10/18: 
  • 17/10/18:
  • 30/10/18: Distribution of the sum of the independent discrete r.v.'s (convolution formula), Bernoulli r.v., Binomial r.v.
Textbook:

Assignment: Prove that, if X and Y are discrete r.v.'s, then Var(X+Y)=Var(X)+Var(Y)
  • 31/10/18: Independent random variables, X,Y independent ==> E[XY]=E[X]E[Y] but the viceversa does not hold, Example with Binomial distribution.
Textbook:

Assignment: Problem 9.32
  • 05/11/18: Geometric r.v., geometric sum and series, Poisson random variable.
  • 06/11/18: Origin of the Poisson distribution (no proof), Poisson approximation of the binomial, cumulative distribution function (cdf), cdf for a discrete random variable, cdf for the geometric distribution.
  • 07/11/18: Memoryless property of the geometric distribution (with proof), uniform distribution, continuous distribution, probability density function.
  • 12/11/18: Density function of a uniform random variable, density functions are not unique, from CDF to densities, interpretation of densities.
  • 13/11/18: