This course introduces probability models and analysis methods on the models to address various phenomenon with uncertainties appropriately. First, the axiomatic probability theory is introduced considering the probability theory that students had already studied at high school. Then, this course formulates random variables and probability distributions. By using these formulations, this course explains how to model various natural and social phenomenon with uncertainties, particularly how to model with the distributions of the linear combination of random variables and the maximum and minimum of random variables. This course also focuses on the stochastic process such as the Poisson process and the Markov process with their applications.
In the analysis of the situation in the management and decision-making, one needs to appropriately address phenomenon with uncertainties. To attain it, it is important to assume a probability model which fits the phenomenon and perform analyses based on the model. This course will provide various models and analysis methods as well as their applications to real problems. Through this course, students will study how to correspond a real problem to a mathematical model and then analyze phenomenon with uncertainties. Some student applications beyond the course will be expected.
By the end of this course, students will be able to:
(1) Explain the difference between the high school probability and the axiomatic probability.
(2) Understand the random variable, the probability distribution and their applications to real problems.
(3) Explain how the distributions of the linear combination of random variables and the maximum and minimum of random variables are used in real problems.
(4) Understand the Poisson process, the Markov process and their applications.
Axiomatic probability, Random variable, Probability distribution, Normal distribution, Law of large numbers, Central limit theorem, Poisson process, Markov process
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
Give a lecture and give exercise problems at the end of class. At the beginning of each class, solutions to the exercise problems are reviewed.
|Course schedule||Required learning|
|Class 1||Probability 1 Sample space, events and the axiomatic probabilty||Understand axiomatic probabilities|
|Class 2||Probability 2 Conditional probability and independence||Calculate conditional distributions and judge independence|
|Class 3||Random variables and probability distributions 1 Discrete probability distributions||Calculate probabilities for discrete probability distributions|
|Class 4||Random variables and probability distributions 2 Continuous probability distributions||Calculate probabilities for continuous probability distributions|
|Class 5||Transformations and characteristics of distributions Expectation and Variance||Calculate expections and variances for various distributions|
|Class 6||Probability generating function and moment generating function Joint distributions||Calculate moments of various distributions with probability generating function or moment generating function|
|Class 7||Independence and covariance Transformations: Bivariate random variables||Transformations of bivariate random variables and derive their distributions|
|Class 8||Summary of the first part of the course and test understanding Solve exercise problems covering the contents of classes 1–7||Test level of understanding and self-evaluate achievement for classes 1–7|
|Class 9||Conditional distributions Conditional expectation||Understand the conditional expectation and calculate it for various distributions|
|Class 10||Distributions of linear combination of random variables Distributions of the maximum and minimum||Derive the distributions of linear combination of random variables, the maximum and the minimum|
|Class 11||Law of large numbers and central limit theorem Markov's inequality and Chebychev's inequality||Evaluate the probability by the Markov's and Chebychev's inequalities|
|Class 12||Poisson process Exponential distribution and Erlang distribution||Calculate probabilities for Poisson process and derive the Erlang distribution|
|Class 13||Renewal process and renewal equation||Derive renewal equations|
|Class 14||Markov chain Classification of states and stationary distributions||Modeling with Markov chain, classification of states and derivation of stationary distributions|
|Class 15||Markov process and Brownian motion||Calculate probabilities for Markov process|
Miyakawa, Masami. Statistical methodology. Tokyo: Kyoritsu Shuppan; ISBN-13: 978-4320016132. (Japanese)
Nothing in particular. Provide handouts when needed.
Midterm and final exams, exercise problems.
Nothing in particular.