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


Last week schedule:

Monday, 9 am - 10 am, Lab ICT 1 (focussed only on group project)

Tuesday, 5 pm - 6 pm, LB08

Wednesday, 3 pm - 4 pm, LB04

Thursday, 11 am - 12 pm, LB120 (tutorial on mock exam, and other questions)


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), solutions (pdf)
  • 20/11/18: doc-cam from 16/10/18 to 19/11/18 (pdf)
  • 23/11/18: doc-cam 20/11/18 (pdf)
  • 28/11/18: exercise on linear regression (pdf)
  • 29/11/18: doc-cam 21/11/18, 27/11/18, 28/11/18 (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: Example: pmf of max and min of two dice, expected value of discrete r.v.'s, E[X+Y]=E[X]+E[Y] with proof (for the discrete case)
Assignment: 9.1, 9.2, 9.3 and 9.14
  • 17/10/18: Expected value of g(X) with X discrete r.v., variance of a discrete r.v.
Assignment: check that Var(X)=E[X^2]-E[X]^2 (alternative definition of variance)
  • 30/10/18: Distribution of the sum of the independent discrete r.v.'s (convolution formula), Bernoulli r.v., Binomial r.v.
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.
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: P[X=a]=0 if X is continuous, expectation of a continuous random variable X, expectation of g(X), Exponential distribution, memoryless property of exponential (with proof), connection between Poisson and exponential (with proof)
  • 14/11/18: Normal distribution, standard Normal distribution, standardisation of a random variable, normal distribution concentrates 95% of its probability mass within 2 standard deviations from the mean.
Assignment: Problems 10.11, 10.36
  • 19/11/18: Joint probability mass function (joint distribution for the discrete case), marginal distributions, probability table, computing probability of events from probability table, checking independence of two random variables from probability table, joint density function (joint distribution for the continuous case).
  • 20/11/18: Interpretation of joint densities, from joint to marginal densities, example: lifetimes of two components modelled with a joint distribution, convolution formula for the density of the sum of two continuous r.v.'s (no proof), independent continuous r.v.'s, E[g(X,Y)] where X,Y have joint density f(x,y), E[X+Y]=E[X]+E[Y] (no proof for the continuous case), if X,Y independent ==> E[XY]=E[X]E[Y] but the viceversa does not hold (no proof for the continuous case).
Assignment: (from the example on lifetimes) check that the marginal density for X is given by (2+x)/6 if x in (0,2) and 0 otherwise.
  • 21/11/18: Covariance and correlation of two random variables, introduction to linear regression, residuals, sum of squared residuals, least-squares regression line.
  • 26/11/18: Lab: work on group project
  • 27/11/18: Least-squares estimates for beta0 and beta1, geometric interpretation of least-squares line, role of correlation, properties of the residuals.
  • 28/11/18: Goodness of fit and coefficient of determination R^2, normal linear regression, maximum likelihood and least-squares estimates, 95% prediction interval.
  • 29/11/18: tutorial: problem on normal linear regression, solutions of exercises 1, 2 and 3 from the exam of May 2018.