2024 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): Yumiharu Nakano / Naoto Miyoshi / Moeko Yajima
Class Format: Lecture/Exercise (Face-to-face)
Media-enhanced courses: -
Day of week/Period (Classrooms): 7-8 Mon / 5-8 Thu
Class: -
Course Code: MCS.T212
Number of credits: 210
Course offered: 2024
Offered quarter: 2Q
Syllabus updated: Mar 17, 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	Know the specific plan of the class and the goals to be achieved.
Class 2	Finite Trials	Understand the theory of finite trials.
Class 3	Probability Spaces, Random Variables	Understand the definition of probability spaces and random variables, as well as their fundamental properties.
Class 4	Exercises regarding the contents covered up to the 3rd lecture	Cultivate more practical understanding by doing exercises.
Class 5	Probability Spaces, Random Variables	Understand the definition of probability spaces and random variables, as well as their fundamental properties.
Class 6	Probability Spaces, Random Variables	Understand the definition of probability spaces and random variables, as well as their fundamental properties.
Class 7	Exercises regarding the contents covered up to the 6th lecture	Cultivate more practical understanding by doing exercises.
Class 8	Expectations	Understand the definition of expectations and their fundamental properties.
Class 9	Expectations	Understand the definition of expectations and their fundamental properties.
Class 10	Exercises regarding the contents covered up to the 9th lecture	Cultivate more practical understanding by doing exercises.
Class 11	Convergence of random variables and expectations	Understand convergences of random variables and expectations.
Class 12	Convergence of random variables and expectations	Understand convergences of random variables and expectations.
Class 13	Exercises regarding the contents covered up to the 12th lecture	Cultivate more practical understanding by doing exercises.
Class 14	Sequences of independent random variables	Understand basic results on sequences of independent random variables.
Class 15	Exercises regarding the contents covered up to the 14th lecture	Cultivate more practical understanding by doing exercises.
Class 16	Sequences of independent random variables	Understand basic results on sequences of independent random variables.
Class 17	Law of Large Numbers	Understand the law of large numbers.
Class 18	Exercises regarding the contents covered up to the 17th lecture	Cultivate more practical understanding by doing exercises.
Class 19	Characteristic Functions, Convergence in Law	Understand the characteristic functions and the convergence in law.
Class 20	Central Limit Theorem	Understand the central limit theorem.
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.

P. Bremaud, An Introduction to Probabilistic Modeling, Springer.
N. Yoshida, Fundamentals of Probability to Statistics, Nippon-Hyoron-Sya. (Japanese)
H. Sato, From Measures to Probability, Kyoritu-Shuppan (Japanese).
K. Ito, Introduction to Probability Theory, Cambridge University Press.
W. Feller, An Introduction to Probability Theory and Its Applications, Wiley.

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).