Course: 2022/2023

Probability

(17753)

Knowledge acquisition of: 1) random variables, elementary probability and distributions; 2) relevant probabilistic inequalities; 3) random vectors, marginal and joint distributions; 4) sequences of random variables and concepts of convergences;

Skills and learning outcomes

Description of contents: programme

1. Random experiments
1.1 Events
1.2 Probability
1.3 Conditional probability
1.4 Bayes' formula
1.5 Independence
1.6 Combinatorics
2. Discrete random Variables
2.1 Definition of random variable
2.2 Probability mass function and cumulative distribution function
2.3 Mean, variance, and quantiles
2.4 Binomial, Geometric, Poisson, Negative Binomial, and Hypergeometric distributions
3. Continuous random variables
3.1 Density mass function and cumulative distribution function
3.2 Mean, variance, and quantiles
3.3 Transformations of a random variable
3.4 Uniform, Exponential, Normal, Gamma, and Beta distributions
4. Random vectors
4.1 Joint distributions, marginal distributions, and conditional distributions
4.2 Independence
4.3 Transformations of random vectors
4.4 Multivariate Normal and Multinomial distributions
4.5 Sums of random variables
4.6 Mixtures
4.7 General concepto of random variable
4.8 Random sample
4.9 Order statistics
5. Properties of the expectation
5.1 Expectations of sums of random variables
5.2 Covariance
5.3 Conditional expectation
5.4 Conditional variance
5.5 Moment generating function
6. Limit Theorems
6.1 Markov and Chebishev inequalities
6.2 Weak Law of Large Numbers (convergence in probability)
6.3 Strong Law of Large Numbers (almost sure convergence)
6.5 Central Limit Theorem (convergence in distribution)

Assessment System

- % end-of-term-examination 50
- % of continuous assessment (assigments, laboratory, practicals...) 50

Basic Bibliography

- Sheldon Ross. A First Course in Probability. Pearson Prentice Hall. 2010

- Sheldon Ross · A First Course in Probability : http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf

Additional Bibliography

- Charles M. Grinstead. Grinstead and Snell's Introduction to Probability. University Press of Florida. 2009
- Dimitri P. Bertsekas, John N.Tsitsiklis. Introduction to Probability. Athena Scientific. 2008

- Charles M. Grinstead · Introduction to Probability : https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf

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The course syllabus may change due academic events or other reasons.

**More information: **https://portal.uc3m.es/portal/page/portal/dpto_estadistica/personal/ana_arribas_gil