MATH-F-105

• Contenu du cours

This is an introductory course to probability theory.
This is the tentative table of contents:

1. Probability spaces.

1.1. Sample space.
1.2. Events.
1.3. sigma-algebra.
1.4. Probabilities.

2. Elementary discrete models.

2.1. Laplace experiments.
2.2. Multinomial and hypergeometric distribution.
2.3. Discrete probability spaces.
2.4. From hypergeometric to binomial distribution.
2.5. Poisson distribution.
2.6. Geometric distribution.

3. Random variables.

3.1. Random variables.
3.2. The Borel $\sigma$-algebra.
3.3. Distribution functions.
3.4. Densities.

4. Moments.

4.1. Expected value of a discrete random variable.
4.2. Examples.
4.3. Some results on convergence of series.
4.4. Properties.
4.5. The general case.
4.6. Variance and higher order moments.
4.7. Moment generating functions.

5. Conditional probabilities and independence.

5.1. Conditional probability.
5.2. Total probability and Bayes' theorem.
5.3. Higher order models.
5.4. Independence of events.
5.5. Independence of random variables.

6. Random vectors.

6.1. Random vectors and sequences.
6.2. Covariance and correlation.

7. The law of large numbers.

7.1. Convergence in Lp and in probability.
7.2. The law of large numbers (LLN).
7.3. Applications.

8. The central limit theorem.

8.1. Convergence in distribution.
8.2. The Central limit theorem.
8.3. Applications.

Exercices (Solutions on Univ. virtuelle)

Ex 1
Ex 2
Ex 3
Ex 4
Ex 5
Ex 6
Ex 7
Ex 8
Ex 9
Ex 10
Ex 11

Sample Exam

Exercises
Theory



Updated: May 16, 2017