2020 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 (Zoom)
Media-enhanced courses: -
Day of week/Period (Classrooms): 7-8 Mon (W833) / 5-8 Thu (W833)
Class: -
Course Code: MCS.T212
Number of credits: 210
Course offered: 2020
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	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	Distribution Functions	Understand the definition of distribution functions and their basic properties.
Class 11	Random Variables	Understand the definition of random variables.
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.

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 textbooks and other course material.

Textbook(s)

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)

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

Prerequisites

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