Welcome, this is the website for the Basic Probability course, during the semester 02-2022. Here you will find:
An update of the course content covered on session-by-session basis.
Bibliography and reading materials.
Test dates and solutions.
Class of November 16th, 2022. From Reference 7, Example 9.30, Theorem 9.31, Theorem 9.32, Theorem 9.33, Theorem 9.34, Definition 9.37, Theorem 9.40, Theorem 9.41.
Class of November 2nd, 2022. From Reference 7, Lemma 9.26, Lemma 9.27, Lemma 9.28, Theorem 9.29.
Class of October 19th, 2022. From Reference 7, Definition 9.22, Example 9.23, Definition 9.24, Lemma 9.25, Lemma 9.26, Lemma 9.27, Lemma 9.28.
Class of October 14th, 2022. From Reference 5, Example page 568, Example page 571 on symmetries of the cube. From Reference 7, Definition 6.1, Definition 6.2, Example 6.3. Problem Set 5 Available.
Class of October 12th, 2022. From Reference 5, Examples page 557 (coloring a pentagon with 3 colors). Section 14.4 Polya's Counting Formula. Example page 561, factorization cycle of D_4, counting number of k-colorings of the square. Theorem 14.3.1: Counting the number of fixed k-colorings by a permutation decomposed in cycles. The cycle index polynomial, page 565, Theorem 14.3.2.
Class of October 7th, 2022. From Reference 6 the Dihedral Group (pages 23-26). From Reference 5, Examples page 557.
Class of October 5th, 2022. from Reference 2. Problem Set 4 available.
Class of September 20th, 2022. from Reference 2. General Introduction on Coloring Problems. Definition 18.1, Example 18.2, Example 18.3, Example 18.4, Definition of Subgroup and Cosets. Proposition 18.6, Corollary 18.7. Permutation groups and the concept of ACTION. Proposition 18.8, Definition 18.9, Definition 18.11, Example 18.12, Proposition 18.4.
Class of September 28th, 2022. from Reference 2. Theorem 8.24, Discussion on Example 8.26, Highlights on Theorem 8.27
Class of September 16th, 2022. from Reference 2. Theorem 8.15, Definition 8.18, Example 8.17, Lemma 8.20, Theorem 8.21.
Class of September 14th, 2022. from Reference 2. Example 8.9, Catalan Numbers, Theorem 8.13.
Class of September 9th, 2022. from Reference 2. Definition8.1, Recurrence Relations: Example of the Exponential Growth, Example of the Fibonacci Numbers. Lemma 8.4 (product of absolutely convergent series). Theorem 8.5, Example 8.6.
Class of September 7th, 2022. from Reference 2. Alternative proof of Proposition 6.18, Example 7.1, Theorem 7.3 with two ways: by induction and using combinatorial arguments. Derrangements, Example 7.4, Theorem 7.5, Theorem 7.6. Problem Set 3 available.
Debate Session September 5th, 2022. From Problem Set 1, problems 16 and 32 for the case n = p^k, with p prime, k a natural number.
Class of September 2nd, 2022. from Reference 2. Theorem 6.14; comments on the bases of the space of polynomials of degree n. Example 6.16, Example 6.17. Lemma 6.15 (the transition lemma). Proposition 6.18 with 2 intermediate results. Lemma 6.19, definition of the sets ODD(n), EVEN(n) within S(n). End of chapter 6 on permutations' cycle structure. Problem Set 2 available.
Class of August 31st, 2022. from Reference 2. Chapter 6, Composition of permutations, Lemma 6.4, Definition 6.5, Corollary 6.6, Example 6.7. Comments on the cycle notation. Comments on Circular Permutations, Theorem 6.9, Example 6.10, Definition 6.11, Theorem 6.12.
Debate Session August 29th, 2022. From Problem Set 1, problems 6, 13, 15, 24, 30.
Class of August 26th, 2022. from Reference 2. Theorem 5.18, Example 5.19, Theorem 5.20, Definition of TYPE for a partition set of [n], Theorem 5.22. Chapter 6, Introduction to cycles within permutations and matrix notation of permutations.
Class of August 24th, 2022. from Reference 2. Definition 5.5, Example 5.6, Theorem 5.8, Corollary 5.9.
Class of August 19th, 2022. from Reference 2. Example 4.10, Definition 4.11, Theorem 4.12, Theorem 4.13, comments on section 4.3. Definition 5.1, Corollary 5.3, Corollary 5.4, Theorem 5.2, Definition 5.5.
Class of August 17th, 2022. From Reference 2. Theorem 4.7, Theorem 4.8, Corollary 4.9. From Reference 3 Proposition 1.35, Theorem 1.36, Theorem 1.37. Debate about conditioning through lines of slope -1.
Class of August 12th, 2022. All topics from Reference 2. Theorem 3.13. Section 3.3, Definition 3.15, Theorem 3.16, Example 3.12, Proposition 3.18, Example 3.19, Theorem 3.21. Theorems 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6. Problem Set 1 Available.
Class of August 10th, 2022. Introduction of the course, its nature, prerequisitis and objectives. All topics from Reference 2. Section 3.1 Permutarions, Definition 3.1, Theorem 3.2. Example 3.4, Definition of Multiset, Theorem 3.5. Section 3.2, Theorem 3.6, Example 3.5, Definition 3.8, Example 3.11. Problem Set 0, Available.