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Probability
Engineering & Social Sciences Program
Madrid, Spain
Dates: 1/16/25 - 6/4/25
Probability
OVERVIEW
CEA CAPA Partner Institution: Universidad Carlos III de Madrid
Location: Madrid, Spain
Primary Subject Area: Mathematics
Instruction in: English
Course Code: 18269
Transcript Source: Partner Institution
Course Details: Level 200
Recommended Semester Credits: 3
Contact Hours: 42
DESCRIPTION
1. Probability and random phenomena.
1.1 Random phenomena, sample space, events.
1.2 Axioms of Probability and elementary properties.
1.3 Conditional probability and independence.
1.4 Total probability rule and Bayes¿ formula.
2. Random variables.
2.1 Definition of random variable.
2.2 Expectation, characteristic features, and moments of a random variable.
2.3 Discrete probability models.
2.4 Continuous probability models.
2.5 Transformations of random variables.
3. Jointly distributed random variables
3.1 Definition of random vector, joint, marginal, and conditional distributions.
3.2 Independent random variables.
3.3 Some multivariate distribution models.
3.4 Transformations.
4. Properties of the expectation.
4.1 Expectations of transformation of random variables.
4.2 Covariance, variance of sums, and correlation.
4.3 Conditional expectation.
4.4 Moment generating functions.
5. Limit Theorems.
5.1 Chebyshev¿s inequality.
5.2 Convergence in probability, the Weak Law of Large Numbers.
5.3 Almost sure convergence, the Strong Law of Large Numbers.
5.4 Convergence in distribution, the Central Limit Theorem.
1.1 Random phenomena, sample space, events.
1.2 Axioms of Probability and elementary properties.
1.3 Conditional probability and independence.
1.4 Total probability rule and Bayes¿ formula.
2. Random variables.
2.1 Definition of random variable.
2.2 Expectation, characteristic features, and moments of a random variable.
2.3 Discrete probability models.
2.4 Continuous probability models.
2.5 Transformations of random variables.
3. Jointly distributed random variables
3.1 Definition of random vector, joint, marginal, and conditional distributions.
3.2 Independent random variables.
3.3 Some multivariate distribution models.
3.4 Transformations.
4. Properties of the expectation.
4.1 Expectations of transformation of random variables.
4.2 Covariance, variance of sums, and correlation.
4.3 Conditional expectation.
4.4 Moment generating functions.
5. Limit Theorems.
5.1 Chebyshev¿s inequality.
5.2 Convergence in probability, the Weak Law of Large Numbers.
5.3 Almost sure convergence, the Strong Law of Large Numbers.
5.4 Convergence in distribution, the Central Limit Theorem.