Basic Probability
WELCOME AND SUMMARY
Welcome, this is the website for the Basic Probability course, during the semester 01-2022. Here you will find:
An update of the course content covered on session-by-session basis.
Bibliography and reading materials.
Test dates and solutions.
BIBLIOGRAPHY
Discrete Probability Models and Methods, 2017. Pierre Brémaud Springer. (Available through the electronic library for UNAL students)
Markov Chains. Pierre Brémaud, Springer (Available through the electronic library for UNAL students)
How to Count: an introduction to Combinatorics, 2nd Ed 2011. Allenby & Slomson.
Introduction to Enumerative and Analytic Combinatorics, 2nd Ed 2015. Miklos Bona.
A Course in Functional Analysis, Second Edition. John B. Conway. (Available through the electronic library for UNAL students)
Lecture Notes in Measure Theory. Jaime San Martín (Free Distribution)
Lecture Notes in Basic Probability Theory. Fernando A Morales (Free Distribution)
LECTURES' LOG
Class of June 22th, 2022. LAST CLASS. All topics exposed from Reference 11. Theorem 5.2.2, Theorem 5.2.3, Theorem 5.2.5, Theorem 5.3.1, Theorem 5.3.3.
Class of June 17th, 2022. All topics exposed from Reference 11. Theorem 5.1.4, Theorem 5.1.5, Definition 5.1.6, Theorem 5.1.8, Corollary 5.1.9, Theorem 5.1.10, Theorem 5.2.1.
Class of June 15th, 2022. All topics exposed from Reference 11. Example 4.5.8, Proposition 4.5.11, Theorem 4.5.12, Theorem 4.5.13, Theorem 4.5.14. Comments on Theorem 4.5.16. Definitions 5.1.1, 5.1.2, Theorem 5.1.3.
Class of June 3th, 2022. All topics exposed from Reference 11. Theorem 4.4.12, Definition 4.5.1, Proposition 4.5.2, Theorem 4.5.4, Definition 4.5.5.
Class of June 1st, 2022. All topics exposed from Reference 1. Definition 2.2.1, Example 2.2.2, Example 2.2.3, Example 2.2.4, Theorem 2.2.5, Theorem 2.2.6. Theorem 2.2.8, Theorem 2.2.9.
May 27th, 2022. All topics exposed from Reference 1. Definition 2.1.43, Example 2.1.44. Theorem 2.1.45, Theorem 2.1.46, Definition 2.1.47. Definition 2.1.48, Example 2.1.49, Definition 2.1.50, Definition 2.1.51
May 25th, 2022. All topics exposed from Reference 1. Lemma 2.1.37, Theorem 2.1.36. Definition 2.1.38, Example 2.1.39, Theorem 2.1.40, Theorem 2.1.41, Example 2.1.42.
May 20th, 2022. All topics exposed from Reference 1. Theorem 2.1.27, Theorem 2.1.28, Definition 2.1.29, Theorem 2.1.30, Theorem 2.1.31, Definition 2.1.33, Theorem 2.1.35.
May 18th, 2022. All topics exposed from Reference 1. Definition 2.1.14, Theorem 2.1.15, Definition 2.1.16, Definition 2.1.19, Definition 2.1.20, Example 2.1.21, Example 2.1.21, Theorem 2.1.24, Theorem 2.1.25, Theorem 2.1.26.
May 13th 2022. All topics exposed from Reference 1. Conditional Probability Definition 1.3.7, Theorem 1.3.8, Theorem 1.3.9, Definition 1.3.14. Random Variables. Definition 2.1.1, Example 2.1.3, Example 2.1.4, Theorem 2.1.5, Theorem 2.1.6, Definition 2.1.7, Example 2.1.8, Example 2.1.10, Theorem 2.1.12, Definition 2.1.13.
May 11th, 2022. All topics exposed from Reference 11. Theorem 2.3.1, Theorem 2.3.6, Corollary 2.3.7, Definition 2.3.8, Theorem 2.3.9, Definition 2.4.1, Example 2.4.2, Definition 2.4.3, Example 2.4.4, Lemma 2.4.5, Theorem 2.4.6, Theorem 2.4.7, Theorem 2.4.8.
May 6th, 2022. All topics exposed from Reference 11. Definition 3.6.1, Example 3.6.2, Definition 3.6.3, 3.6.4, Remark 3.6.5, Definiton 3.2.2, Theorem 3.2.3, Definition 2.3.4, Example 2.3.5, Theorem 3.6.6, Remark 3.6.7.
May 4th, 2022. All topics exposed from Reference 11. Theorem 3.1.6, Theorem 3.1.7, Theorem 3.5.2, Example 3.5.3, Theorem 3.5.4. Comments on double sums: Remark 3.5.1. Comments on Product Measure: Remark 3.5.5. Comments on Riemann Integrals and Interated Riemann Integrals in R^2: Hypothesis 3.5.6, Remark 3.5.7, Theorem 3.5.8, Remark 3.5.9. Comments on Uniform Continuity and its role proving that continuous functions on closed bounded intervals are Riemann Integrable: Definition 3.5.10, Example 3.5.11.
April 29th, 2022. Theorem 3.2.1, Lemma 3.3.1, Theorem 3.4.1. Examples.
April 27th, 2022. All topics exposed from Reference 11. Theorem 2.2.7, Theorem 1.1.41, Theorem 2.2.8. Theorem 2.2.9. Definition 3.0.1, Example 3.0.2, Theorem 3.1.1, Remark 3.1.2, Definition 3.1.4, Example 3.1.5
April 22nd, 2022. All topics exposed from Reference 11. Section 1.2.4 Closing Remarks on Hilbert Spaces. Discussion on measure, motivations and aims. Illustration of Riemann Integral vs Lebesgue Integral paradigms, example of indicator function of the irrational numbers in [0,1]. Section 2.1, Definition 2.1.1, Definition 2.1.2, Example 2.1.3. Section 2.2, Definition 2.1.1, Examples 2.1.2, 2.1.3, 2.1.4, 2.1.5, 2.1.6, Theorem 2.2.7, Theorem 1.1.41, Theorem 2.2.8.
April 20th, 2022. All topics exposed from Reference 11. Theorem 1.2.21 (The Projection Theorem on Subspaces), Theorem 1.2.23 (Properties of the Projection Operator), Definition 1.2.15 Orthogonal Complement. Section 1.2.3, The Riesz Representation Theorem: Definition 1.2.24 (Linear Functionals), Example 1.2.25, Example 1.2.26, Theorem 1.2.27 (Characterization of Continuous Linear Functionals), Example 1.2.28, Theorem 1.2.31 (Riesz Representation Theorem, incomplete version).
April 6th, 2022. All topics from Reference 9: Cauchy-Shwartz-Bunyakovsky inequality, Corollary I.1.5, Example 1.8. Section I.2 Orthogonality, Definition I.1.1, Theorem I.2.2, Theorem I.2.3, Definition I.2.4, Theorem I.2.5.
April 1st, 2022. Topics from Reference 8: Corolario 2, 3, 4 pg 140, Teorema 19, Corolario 1, 2, Teorema 22. Topics from Reference 9: Definition I.1.1, Example I.1.2, Section I.1.4, Definition I.1.6, Example I.1.7
March 30th, 2022. All topics from Reference 8. Cauchy Sequences, Section IV.5: Teorema 12, Lema 1, Lema 2, Teorema 13. Section IV.6: Definition of Limits to Infinity. Section IV.7 Series: Teorema 15, Definition of Conditional and Absolute Convergence, Teorema 18, Corolario 2, 3, 4.
March 25th, 2022. All topics from Reference 8. Axiom of the supremum (pg 76). Sequences, bounded sequences, Subsequences, Monotone sequences (Section IV.1). Limit of a sequence, (Section IV.2), Teorema 1, Teorema 2, Teorema 3, Teorema 4. Definition of Superior and Inferior Limits, limsup, liminf (pg 122). Corolario 1 (pg 123).
March 23rd, 2022. Cycle structure of permutations, Section 4.2 and Theorem 4.27 from Reference 7. Examples of probabilistic questions linked to cycles of permutations.
March 18th, 2022. Application of the Inclusion-Exclusion Principle: Derrangements, Section 2.4.3.4 from Reference 7. Multisets, section 2.1.1 from Reference 7. Counting permutations with order constraints.
March 16th, 2022. Closing remarks on the paths approach to combinatorial numbers. Comments on conditioning, examples structuring the power set of [n]. The Multinomial Theorem, inquiry-based approach. The natural arise of multinomial coefficients, Theorem 1.41 from Reference 7 and the weak compositions of n in k parts. Introduction of strong compositions of n in k parts as an intermediat step. Counting strong compositions Corollary 2.5 from Reference 7, counting weak compositions Corollary 2.3 from Reference 7. Comments on additional bijections between weak and strong compositions. The inclusion-exclusion principle: indicator functions notation, Theorem 2.38 from Reference 7.
March 11th, 2022. All results from Reference 7. The Product Principle: Theorems 1.6, 1.8 and Corollary 1.11. Permutations: Theorems 1.15 and 1.17. Circular permutations: Example 1.22. Combintions/Counting k-subsets: Theorem 1.23. The Binomial Theorem 1.24. Bijections as counting tools: Definition 1.27, Corollary 1.28. Properties of the Binomial Coefficients Proposition 1.35, Theorem 1.36, Theorem 1.37.
March 9th, 2022. Presentation of the course, evaluation system, aims and expectations. Lecture exctracted from Introduction in Reference 2. Some philosophical remarks on the meaning of probability, rather than its computation. What can be known about a random process, despite it uncertainty? For the following basic examples we surfed through its meaning a probability distribution function or scaled histogram:
Tossing one coin.
Throwing one die. Throwing a biased/unfair die.
Sum of two thrown dies.
Tossing a coin three times.