# ST2004_17_18

**ST2004 - Applied Probability I **

**2017/2018**

Module Code: ST2004

Module Title: Applied Probability I (module descriptor)

Credit Value: 5

Cohorts: MSISS, CSL(JS)

Week range: 05-10, 12-16

Semester: S1 (Michaelmas)

Total hours: 33

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

Additional material will be provided when needed.

*Syllabus (lecture by lecture):*

26/09/17: Monte Carlo approach, Empirical Law of Large Numbers, Birthday problem (via both simulation and combinatorics).

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

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27/09/17: True and pseudo random number generation, Multiplicative congruential method, Random generation from an interval, Random generation from integers, Generation of random permutations.

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

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02/10/17: Lab 1 - generating random passwords.

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03/10/17: More on generation of random permutations, 1970 draft lottery problem.

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

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04/10/17: Frequentist probability, Axiomatic foundations of probability.

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

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09/10/17: Lab 2 - does skill win the league?

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10/10/17: 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 want them to be checked -not marked- hand your solutions in on Wednesday 18/10/17).

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12/10/17: Probability on an uncountable sample space (dartboard example), Derivation of basic rules of probability from axioms, Independent events.

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

Assignment: Probl. 7.27 and 7.28 (if you want them to be checked -not marked- hand your solutions in on Wednesday 18/10/17).

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17/10/17: Monty Hall problem via simulation, Conditional probability, Chain rule, Bayes rule.

Textbook: Chapter 6 (pp. 212-217, 6.1.3 excluded, pp. 220-222), Chapter 8 (pp. 256-258, pp. 260-261).

Assignment: Problems 8.1-8.3 (if you want them to be checked -not marked- hand your solutions in on Tuesday 31/10/17).

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18/10/17: Tutorial: solution of Exercise 7.3, 7.4, 7.5 and 7.14. Binomial coefficient.

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23/10/17: Lab 3 - system lifetime.

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24/10/17: Law of conditional probability (or law of total probabilities), Bayes theorem, Monty hall problem with Bayes theorem, Problems 8.30 and 8.19.

Textbook: pp. 264-267.

Assignment: 8.17, 8.31, 8.32 (if you want them to be checked -not marked- hand your solutions in on Tuesday 31/10/17).

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25/10/17: Random variables (RV), Range of a RV, Probability mass function (or distribution) of a discrete RV, Expected value of a discrete RV.

Textbook: Chapter 9, pp. 283-290.

Assignment: 9.1, 9.2, 9.3 (if you want them to be checked -not marked- hand your solutions in on Tuesday 31/10/17)

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31/10/17: Linearity of expectation: E[X+Y]=E[X]+E[Y] (proof for discrete R.V.s); Expectation of g(X); Counterexample disproving E[g(X)]=g(E[X]); E[aX+b]=aE[X]+b.

Textbook: pp. 290-294.

Assignment: 9.14.

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01/11/17: Measures of dispersion: variance and standard deviation of a R.V.; Equivalent definitions for the variance; Independent R.V.s (general case);

Textbook: pp. 294-295, p. 299.

Assignment: prove that Var(aX+b)=a^2Var(X) (prove it for X discrete).

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13/11/17: Lab 4 - Summaries of variation.

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14/11/17: Independent R.V.s (discrete case); X and Y independent => E[XY]=E[X]E[Y] (but not viceversa); Distribution of sum of independent R.V.s (discrete case); Bernoulli distribution.

Textbook: pp. 300-301, convolution formula at p. 302, 303-304.

Assigment:

- Toss of two dice: let X=sum of outcomes, Y=max of outcomes, and prove that X and Y are not independent.

- Prove that if X and Y are independent then Var(X+Y)=Var(X)+Var(Y).

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15/11/17: Binomial distribution; Geometric distribution.

Textbook: pp. 304-305; Section 9.6.5. at p. 312.

Assignment: Problem 9.32.

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20/11/17: Geometric sum and geometric series; Poisson distribution; Poisson approximation of the binomial distribution (no proof).

Textbook: pp. 111-113;

Assignment: Verify that if X is Geometric(p), the sum for k that goes from 1 to infinity of P(X=k) is equal to 1.

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21/11/17: Cumulative distribution function; Uniform distribution; Continuous random variables; Probability density function

Textbook: pp. 318-322, pp. 333-334.

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22/11/17: Density function is not unique; From cumulative distribution function to density; Interpretation of density functions.

Textbook: pp. 322-326.

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27/11/17: If X is continuous, then P(X=x)=0 for every x; Exponential distribution; Exponential r.v. as transformation of a uniform r.v.; inverse-transform method.

Textbook: pp. 335-336, pp. 353-354.

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28/11/17: Expectation for a continuous r.v. X; Expectation of g(X); Memoryless property of the exponential distribution; Connection between exponential and Poisson; Normal distribution.

Textbook: pp. 326-330, pp. 336-339, pp. 343-344.

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29/11/17: Standard normal distribution; Standardisation; Joint probability mass function of two discrete r.v.'s X and Y; Marginal distributions; Probability table.

Textbook: pp. 344-345, pp. 360-362.

Assignment: Problems 10.11, 10.36.

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30/11/17: Joint density of two continuous r.v.'s X and Y; Interpretation of the joint density; Marginal densities; Convolution formula (no proof).

Textbook: pp. 362-368.

Assignment: Problem 11.12.

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06/12/17: Tutorial by Fearghal Donaghy. Solution to mock exam; Covariance of two random variables; If X and Y are independent => Cov(X,Y)=0 but not viceversa.

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11/12/17: Independent random variables (continuous case); Expected value of g(X,Y); X and Y continuous independent RVs => E[XY]=E[X]E[Y]; X and Y continuous RVs => E[X+Y]=E[X]+E[Y]; Simple linear regression.

Textbook: pp. 367-374

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12/12/17: Least squares method: derivation of optimal regression coefficients. Geometric interpretation of least squares line.

Textbook: /

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13/12/17: Properties of residuals: null mean and zero correlation with the predictor; Decomposition of variance; coefficient of determination.

Textbook: /

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14/12/17: Normal linear regression; 95% prediction intervals for the response variable.

Textbook: /

Exercise on regression