Course: 2024/2025

Probability

(17753)

The course Objectives are:
1. Understand the fundamental concepts of probability and random experiments.
2. Analyze events and calculate probabilities using various techniques.
3. Explore conditional probability and apply Bayes' formula.
4. Recognize independence in random events and perform combinatorial analysis.
5. Define discrete random variables and analyze their properties.
6. Examine different discrete probability distributions (Binomial, Geometric, Poisson, etc.).
7. Introduce continuous random variables and study their characteristics.
8. Analyze continuous probability distributions (Uniform, Exponential, Normal, etc.).
9. Understand and work with random vectors, including joint, marginal, and conditional distributions.
10. Investigate properties of random vectors, including independence and transformations.
11. Explore the concepts of sums, mixtures, and random samples.
12. Analyze the concept of order statistics in random samples.
13. Study the properties of expectation, covariance, conditional expectation, and variance.
14. Examine moment generating functions.
15. Explore limit theorems such as Markov and Chebyshev inequalities.
16. Understand convergence in probability, almost sure convergence, and convergence in distribution.
17. Apply the Central Limit Theorem to analyze the behavior of sample means.

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)

Learning activities and methodology

TRAINING ACTIVITIES OF THE STUDIES PROGRAM
AF1 Theory class
AF2 Exercises class
AF4 Lab class
AF5 Office hours
AF6 Group office hours
AF7 Individual student's work
AF8 In-class assessment activities
Activity Code Total nb hours Total in-class nb hours % in-class hours
AF1 33 33 100
AF2 15 15 100
AF4 15 15 100
AF5 12 12 100
AF6 30 0 0
AF7 115,5 0 0
AF8 4,5 4,5 100
COURSE TOTAL 225 75 33
TEACHING METHODOLOGIES USED IN THIS COURSE
MD1 Teacher class presentations with visual support to present the main contents of the course.
MD3 Exercises and case studies to be solved individually or in group

Assessment System

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

Calendar of Continuous assessment

Basic Bibliography

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

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://math.dartmouth.edu/~prob/prob/prob.pdf

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