### 2019　Fundamentals of Probability

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Miyoshi Naoto  Nakano Yumiharu
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Mon7-8(W833)  Thr5-8(W833)
Group
-
Course number
MCS.T212
Credits
3
2019
Offered quarter
2Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

This course emphasizes that students learn the basic skills of probabilistic representation of random phenomena and gives lectures on fundamental concepts of probability theory. The course also facilitates students' understanding by giving exercises and assignments.

### Student learning outcomes

Students will be able to acquire the basic skills of mathematical representation for probabilistic phenomena.

### Keywords

Probability space, Independence and conditional probability, Random variables and their distributions, Lows of large numbers, Central limit theorem

### Competencies that will be developed

 Intercultural skills Communication skills ✔ Specialist skills Critical thinking skills ✔ Practical and/or problem-solving skills

### Class flow

Two 90 minute lectures and one 90 minute exercise per week.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Introduction to Probability Understand the necessity for the idea of probability.
Class 2 Set Operations and sigma-fields Understand the definition of sigma-fields
Class 3 Exercises regarding the contents covered up to the 2nd lecture Cultivate more practical understanding by doing exercises.
Class 4 Probability Spaces and Basic Properties of Probability Understand the definition of probability spaces and their basic properties.
Class 5 Limits of Event Sequences and Continuity of Probability Understand the limits of event sequences and the concept of continuity of probability.
Class 6 Exercises regarding the contents covered up to the 5th lecture Cultivate more practical understanding by doing exercises.
Class 7 Conditional Probability Understand the definition of conditional probability and solve the related basic exercise problems.
Class 8 Independence of Events Understand the concept of independence of probabilistic events.
Class 9 Exercises regarding the contents covered up to the 8th lecture Cultivate more practical understanding by doing exercises.
Class 10 Random Variables Understand the definition of random variables.
Class 11 Distribution Functions Understand the definition of distribution functions and their basic properties.
Class 12 Exercises regarding the contents covered up to the 11th lecture Cultivate more practical understanding by doing exercises.
Class 13 Independent Random Variables and Convolutions Understand the independence of random variables and the idea of convolutions.
Class 14 Expectations Understand the definition of expectations.
Class 15 Exercises regarding the contents covered up to the 14th lecture Cultivate more practical understanding by doing exercises.
Class 16 Variances and Covariances Understand the definition of variances and covariances.
Class 17 Generating Functions and Related Functions Understand the definitions of generating functions, moment generating functions, Laplace transforms and characteristic functions.
Class 18 Exercises regarding the contents covered up to the 17th lecture Cultivate more practical understanding by doing exercises.
Class 19 Convergences of Sequences of Random Variables Understand the convergences of random variables such as almost sure convergence, convergence in probability and convergence in law.
Class 20 Laws of Large Numbers Understand the weak and strong laws of large numbers.
Class 21 Exercises regarding the contents covered up to the 20th lecture Cultivate more practical understanding by doing exercises.
Class 22 Central Limit Theorem and Normal Approximation Understand the central limit theorem and its application to normal approximation.

Not required.

### Reference books, course materials, etc.

Takahashi, Yukio. Probability Theory. Asakura-Shoten. (Japanese)
Nishio, Makiko. Probability Theory. Jikkyo-Shuppan. (Japanese)
Ito, Kiyoshi. Probability Theory. Iwakura-Shoten. (Japanese)

### Assessment criteria and methods

Scores are based on final exam, exercise problems and assignments.

### Related courses

• MCS.T312 ： Markov Analysis
• MCS.T333 ： Information Theory
• MCS.T223 ： Mathematical Statistics

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites.