# ST2004_ST2352

**ST2004 - ST2352 Applied Probability I **

**A.Y. 2016/2017**

Module Code: ST2004 or ST2352

Module Title: Applied Probability I

Cohorts: MSISS, BAI, CSL(JS), Maths, TSM

Week range: 05-10, 12-16

Semester: S1 (Michaelmas)

Total hours: 33

Timetable:

- Monday, h. 9-10, (ICTLAB & ITCLAB2), **room LB04 starting on 28/11**

- Tuesday, h. 17-18, room LB08

- Wednesday, h. 15-16, room M17

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

**SYLLABUS (lecture by lecture):**

*27/09/16:* Monte Carlo approach, Empirical Law of Large Numbers, Birthday problem.

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

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*28/09/16:* 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).

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*3/10/16:* Lab - Generating random passwords.

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*4/10/16: *Generation of random permutations, 1970 draft lottery problem.

Textbook: Section 2.9.4 (pp. 62-63), Section 3.7 (pp. 98-102).

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*5/10/16: *More on 1970 draft lottery, Frequentist probability, Axiomatic foundations of probability.

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

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10/10/16: Lab - Does skill win the league?

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11/10/16: Probabilities on a finite or countable sample space, Equally likely outcomes, Buffon needle problem.

Textbook: Section 7.1.1 (pp. 232-235). Assignment: Problems 7.1-7.5 and 7.14.

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12/10/16: Derivation of basic rules of probability from axioms, Independent events.

Textbook: Section 7.2 (pp. 239-241, 7.2.1 excluded), Section 7.3 (pp. 243-244). Assignment: Probl. 7.27 and 7.28.

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17/10/16: Lab - System lifetime.

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18/10/16: Monty Hall problem, Conditional probability, Chain rule, Bayes rule, Law of conditional probability (or Law of total probabilities)

Textbook: Chapter 6 (pp. 212-127, 6.1.3 excluded, pp. 220-222), Chapter 8 (pp. 256-258, pp. 260-261, pp.264-267). Assignment: Problems 8.1-8.3, 8.15.

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19/10/16: Tutorial - Solution of some problems given as assignment.

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24/10/16: Lab - Summaries of variation.

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25/10/16: Bayes theorem, Monty Hall problem using Bayes theorem, Problems 8.19, 8.30, Random variables.

Textbook: Chapter 9 (pp. 283-285). Assignment: Problems 8.17, 8.31, 8.32.

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26/10/16: Discrete random variables: Probability mass function, Expected value, E[X+Y]=E[X]+E[Y].

Textbook: Chapter 9 (pp. 286-291). Assignment: Problems 9.1, 9.3.

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1/11/16: Expected value of g(X), E[aX+b]=aE[X]+b, Variance and Standard Deviation of a R.V.

Textbook: pp. 292-294. Assignment: Problem 9.14.

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2/11/16: Independent random variables (general and discrete case), E[XY]=E[X]E[Y] if X and Y are independent, Problem 9.25.

Textbook: pp. 299-300. Assignment: Prove that Var(aX+b)=a^2Var(X), Problem 9.19.

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14/11/16: Lab - Simple queues.

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15/11/16: Distribution of the sum of two independent discrete RVs, Bernoulli distribution, Binomial distribution.

Textbook: pp. 300-305, pp. 312-313 (9.6.5). Assignment: Problem 9.32, Prove that Var(X+Y)=Var(X)+Var(Y) if X and Y are independent.

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16/11/16: Tutorial: Solution of some problems given as assignment (8.12, 8.31, 9.14).

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21/11/16: Lab - Group project.

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22/11/16: Geometric distribution, Geometric sum and series, Poisson distribution.

Textbook: pp. 112-115, p. 306, pp. 312-313 (9.6.5).

Assignment: Verify that the probability mass function of a geometric r.v. sums up to 1, Example 9.8.

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23/11/16: Poisson distribution as a limit of Binomial, Cumulative distribution function.

Textbook: pp. 306-307.

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28/11/16: Continuous distributions, Density function, Density function from cumulative distribution function.

Textbook: pp. 318-322. Assignment: Example 10.3, Problem 10.1.

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29/11/16: Interpretation of densities, Exponential distribution, Inverse-transform method.

Textbook: pp. 322-324, pp. 353-354.

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30/11/16: Tutorial by Gernot Roetzer. Problem 10.3 and 10.7, Example 10.5 in the textbook.

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5/12/16: Proof of inverse-transform method, Expectation and Variance of continuous r.v., Mean of exp. r.v., Memoryless property of exp. r.v.

Textbook:pp. 326-330, pp. 335-337. Assignment: Show that Var(X)=1/lambda^2 if X is Exp(lambda), Problem 10.11.

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6/12/16: Connection between Exponential and Poisson, Gamma distribution, Expected value, Connection between Gamma and Poisson.

Textbook: pp. 338-342,. Assignment: Show that Var(X)=a/lambda^2 if X is Gamma(a,lambda).

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7/12/16: Normal distribution, Markov inequality, Chebyschev inequality.

Textbook: pp. 343-345, pp. 448-449. Assignment: Problem 10.36.

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12/12/16: Examples on Chebyshev inequality, Joint probability mass function, Marginal distributions.

Textbook: pp. 360-362. Assignment: Problems 11.1 and 11.3.

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13/12/16: Equivalent definition of independence of discrete r.v.'s, Joint density function for continuous r.v.'s, Interpretation of joint density, Convolution formula for the density of a sum of continuous r.v.'s.

Textbook: pp. 362-365. Assignment: Prove the two definitions (in terms of c.d.f. and in terms of p.m.f.) of independence of two discrete R.V.'s are equivalent.

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14/12/16: Marginal density functions, Equivalent definition of independence of continuous r.v.s, Expected value of g(X,Y), Variance of X+Y, Covariance of (X,Y).

Textbook: pp. 367-374.

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