2021 Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science

Fundamentals of Probability

Academic unit or major: Undergraduate major in Mathematical and Computing Science
Instructor(s): Naoto Miyoshi / Yumiharu Nakano / Moeko Yajima
Class Format: Lecture/Exercise
Media-enhanced courses: -
Day of week/Period (Classrooms): 7-8 Mon (S223) / 5-8 Thu (S223)
Class: -
Course Code: MCS.T212
Number of credits: 210
Course offered: 2021
Offered quarter: 2Q
Syllabus updated: Jul 10, 2025
Language: Japanese

Syllabus

Course overview and goals

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.

Course description and aims

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

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/Objectives

	Course schedule	Objectives
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.

Study advice (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.

Textbook(s)

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)

Evaluation methods and criteria

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

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