Difference between revisions of "MAT2313"

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Foundations of Mathematics (3-0) 3 Credit Hours  
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= Combinatorics and Probability - MAT2313=
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==Course Catalog==  
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''Corequisite'': [[MAT1224]].
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''Content'': Basic counting principles. Permutations and combinations. Binomial and multinomial coefficients. Pigeonhole and inclusion-exclusion principles. Graphs, colorings, planarity. Eulerian and Hamiltonian graphs. Recurrence relations. Generating functions. Prerequisites: MAT1224 Calculus II and MAT 1313 Algebra and Number Systems. 3 Credit Hours  
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''Sample textbooks'': Alan Tucker, Applied Combinatorics (6th ed). Wiley (2012).
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==Topics List==
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Course outline:
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Week 1: Finite sets, strings, enumeration, the addition and product rules.
  
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''Corequisite'': [[MAT1224]].  
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Week 2: Combinations, permutations.
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Week 3: Binomial and multinomial coefficients.
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Week 4: The Pigeonhole Principle. The Inclusion-Exclusion Formula, derangements, the Euler ɸ function (totient).
  
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''Content'': Permutations, combinations, multinational coefficients, inclusion/exclusion principle, axioms of probability, conditional probability, Bayes formula, independent events, discrete random variables, expected value,m variance, discrete random variables (Bernoulli, Binomial, Poisson, geometric, hypergeometric and Zeta random variables), continuous random variables (uniform, normal and other distributions), joint distributions, properties of expectations, limit theorems (Chebyshev's inequality, Central Limit Theorem, Law of Large Numbers)) Generally offered: Fall, Spring, Summer.  
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Week 5: Review. First midterm exam.
  
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==Description==
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Week 6: Graphs and multigraphs.
  
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Introduction to the theory of probability, through the study of discrete and continuous random variables.
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Week 7: Eulerian and Hamiltonian graphs.  
  
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==Sample textbooks==
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Week 8: Trees. Colorings. Planarity.
  
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* Modern Mathematical Statistics with Applications (Springer Texts in Statistics). Jay L. Devore and Kenneth N. Berk. Second Edition.
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Week 9: Review. Second midterm exam.
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* A ''Probability Course for the Actuaries: A Preparation for Exam P/1'', by Marcel B. Finan. Freely available [https://people.cas.uab.edu/~pjung/teaching_files/ProbabilityForActuaries.pdf online].
 
  
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==Topics List==
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Week 10: Generating functions. The Binomial Theorem. Partitions.
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{| class="wikitable sortable"
 
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! Week !! Topic !! Sections from Finan's book !! Subtopics
 
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|  1-2 
 
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|| [[Populations and Samples]]
 
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|| Chapters 1-2
 
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* The Fundamental Principle of Counting
 
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* Permutations and Combinations
 
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* Permutations and Combinations with Indistinguishable Objects
 
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|  3-4 
 
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|| [[Probability: Definitions and Properties]]
 
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|| Chapter 3
 
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* Basic Definitions and Axioms of Probability
 
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* Properties of Probability
 
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* Probability and Counting Techniques
 
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|-
 
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|  5 
 
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|| [[Conditional Probability and Independence]]
 
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|| Chapter 4
 
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* Conditional Probabilities
 
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* Posterior Probabilities: Bayes’ Formula
 
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* Independent Events
 
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* Odds and Conditional Probability
 
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|  6-8 
 
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|| [[Discrete Random Variables]]
 
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|| Chapter 5
 
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* Random Variables
 
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* Probability Mass Function and Cumulative Distribution Function119
 
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* Expected Value of a Discrete Random Variable
 
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* Expected Value of a Function of a Discrete Random Variable
 
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* Variance and Standard Deviation
 
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* Binomial and Multinomial Random Variables
 
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* Poisson Random Variable
 
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* Other Discrete Random Variables (Geometric, Hypergeometric, etc)
 
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|  9-10 
 
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|| [[Continuous Random Variables]]
 
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|| Chapter 6
 
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* Distribution Functions
 
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* Expectation, Variance and Standard Deviation
 
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* The Uniform Distribution Function
 
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* Normal Random Variables
 
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* Exponential Random Variables
 
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* Gamma and Beta Distributions
 
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* The Distribution of a Function of a Random Variable
 
  
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Week 11: Recurrence relations. Linear recurrences.  
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|  11  
 
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|| [[Joint Distributions]]
 
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|| Chapter 7
 
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* Jointly Distributed Random Variables
 
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* Independent Random Variables
 
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* Sum of Two Independent Random Variables
 
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|  12 
 
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|| [[Properties of Expectation]]
 
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|| Chapter 8
 
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* Expected Value of a Function of Two Random Variables . . . . . 351
 
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* Covariance, Variance of Sums, and Correlations . . . . . . . . . 362
 
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* Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 376
 
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* Moment Generating Functions
 
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|  13-14 
 
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|| [[Limit Theorems]]
 
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|| Chapter 9
 
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* The Law of Large Numbers
 
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* The Central Limit Theorem
 
  
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|}
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Week 12: Solving recurrences by generating functions.
  
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==See also==
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Week 13: Exponential generating functions. Nonlinear recurrences.
  
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* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]
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Week 15: Review.

Latest revision as of 16:49, 14 August 2023

Combinatorics and Probability - MAT2313

Corequisite: MAT1224.

Content: Basic counting principles. Permutations and combinations. Binomial and multinomial coefficients. Pigeonhole and inclusion-exclusion principles. Graphs, colorings, planarity. Eulerian and Hamiltonian graphs. Recurrence relations. Generating functions. Prerequisites: MAT1224 Calculus II and MAT 1313 Algebra and Number Systems. 3 Credit Hours

Sample textbooks: Alan Tucker, Applied Combinatorics (6th ed). Wiley (2012).

Topics List

Course outline:

Week 1: Finite sets, strings, enumeration, the addition and product rules.

Week 2: Combinations, permutations.

Week 3: Binomial and multinomial coefficients.

Week 4: The Pigeonhole Principle. The Inclusion-Exclusion Formula, derangements, the Euler ɸ function (totient).

Week 5: Review. First midterm exam.

Week 6: Graphs and multigraphs.

Week 7: Eulerian and Hamiltonian graphs.

Week 8: Trees. Colorings. Planarity.

Week 9: Review. Second midterm exam.

Week 10: Generating functions. The Binomial Theorem. Partitions.

Week 11: Recurrence relations. Linear recurrences.

Week 12: Solving recurrences by generating functions.

Week 13: Exponential generating functions. Nonlinear recurrences.

Week 15: Review.

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