### 2021　Fundamentals of Probability

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Miyoshi Naoto  Nakano Yumiharu  Yajima Moeko
Class Format
Lecture / Exercise
Media-enhanced courses
Day/Period(Room No.)
Mon7-8(S223)  Thr5-8(S223)
Group
-
Course number
MCS.T212
Credits
3
2021
Offered quarter
2Q
Syllabus updated
2021/6/9
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

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

### Class flow

Two 100 minute lectures and one 100 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 Sigma-fields and measurable spaces Understand the definition of sigma-fields and measurable spaces.
Class 3 Exercises regarding the contents covered up to the 2nd lecture Cultivate more practical understanding by doing exercises.
Class 4 Probability Spaces and Fundamental Properties of Probability Understand the definition of probability spaces and their fundamental properties.
Class 5 Conditional Probability and Independence of Events Understand the definition of conditional probability and the notion of independence of probabilistic events.
Class 6 Exercises regarding the contents covered up to the 5th lecture Cultivate more practical understanding by doing exercises.
Class 7 Continuous Probability Distributions and Probability Distribution Functions Understand probability distribution functions and the notion of absolute continuity.
Class 8 Random Variables and Measurable Functions Understand the definitions of random variables and measurable functions.
Class 9 Exercises regarding the contents covered up to the 8th lecture Cultivate more practical understanding by doing exercises.
Class 10 Probability Distributions of Random Variables and Their Convergences Understand the probability distributions of random variables and the notion of their convergences.
Class 11 Expectations Understand the definition of expectations.
Class 12 Exercises regarding the contents covered up to the 11th lecture Cultivate more practical understanding by doing exercises.
Class 13 Variances, Covariances and Moments Understand the definitions of variances, covariances, and moments.
Class 14 Convergence Theorems for Expectations Understand some convergence theorems for expectations.
Class 15 Exercises regarding the contents covered up to the 14th lecture Cultivate more practical understanding by doing exercises.
Class 16 Probability Generating Functions and Moment Generating Functions Understand the definition of probability generating functions and moment generating functions.
Class 17 Characteristic Functions Understand the definitions of characteristic functions.
Class 18 Exercises regarding the contents covered up to the 17th lecture Cultivate more practical understanding by doing exercises.
Class 19 Law of Large Numbers and Central Limit Theorem Understand the law of large numbers and the central limit theorem.
Class 20 Large Deviation Princeple Understand the large deviation principle.
Class 21 Exercises regarding the contents covered up to the 20th lecture Cultivate more practical understanding by doing exercises.
Class 22 Final Exam. Check the level of understanding through the final exam.

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to reference books and other course material.

Not required.

### Reference books, course materials, etc.

Nishio, Makiko. Probability Theory. Jikkyo-Shuppan. (Japanese)
Ito, Kiyoshi. Fundamentals of Probability Theory. Iwakura-Shoten. (Japanese)
Shiga, Tokuzo. From Lebesgue Integrals to Probability Theory. Kyoritsu-Shuppan. (Japanese)
Kumagaya, Takashi. Probability Theory. Kyoritsu-Shuppan. (Japanese)
Takahashi, Yukio. Probability Theory. Asakura-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
• XCO.B103 ： Foundations of Computing 3
• MCS.T332 ： Data Analysis
• MCS.T304 ： Lebesgue Interation

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

No prerequisites, but it is preferable to study Foundations of Computing 3 (XCO.B103).