## Course Title : Probability and Statistics

Code 90280 3rd year 1st term 2 No Restriction Lecture Toshiyuki TANAKA

### Course Description

After summarizing basics of probability theory and statistics, various concepts and methods of modern statistics on the basis of probability theory and statistics, in particular multivariate regression analysis and statistical hypothesis testing, are described. Applications of these concepts and methods to data analysis are also reviewed.

### Grading

Grading is done on the basis of contents of reports submitted and results of written end-term examination.

### Course Goals

Course goals are to master basics of probability theory and statistics, and to understand how statistical methods such as multivariate regression analysis and principal component analysis are used in practice, including their theoretical backgrounds. Deepening understanding of practical applications is also expected.

### Course Topics

Theme Class number of times Description
Basics of probability theory and statistics 4 Following items are described.
In probability theory: Probability space, density functions, characteristic functions, expectation, covariance, correlation coefficient, Gaussian distribution, chi-squared distribution, transformation of random variables, multivariate Gaussian distribution, central limit theorem, law of large numbers.
In statistics: Procedures of statistical testing, estimation of mean and variance, test on mean, test on variance, test on ratio of variances.
Multivariate regression analysis, principal component analysis 4 Describes mean-squared error estimation of regression coefficients in multivariate regression, tests on regression coefficients and regression formula, and partial correlation coefficients. Principal component analysis and its applications are also described.
Statistical testing, parameter estimation 4 Describes likelihood ratio test derived from Bayesian framework and Nayman-Pearson lemma under the framework of statistical decision theory and reviews properties of operating characteristic curve and uniformly most powerful test.
Also describes maximum likelihood estimation and Bayesian estimation for parameter estimation methods.
Statistical learning theory, data analysis 3 Describes statistical learning theory, which is important as a basis for modern applications of statistics to various field. Also reviews practical applications to problems of data analysis.

### Textbook

Printed materials are distributed if appropriate.

### Textbook(supplemental)

C. M. Bishop: Pattern Analysis and Machine Learning, Cambridge University Press. T. Hastie, R. Tibshirani, and J. Friedman: The Elements of Statistical Learning, Springer.   ### Prerequisite(s)

Students are expected to have taken Probability Theory, Mathematical Statistics, Linear Algebra A, and Linear Algebra B in the Liberal Arts and General Education Courses.

### Additional Information

Course topics would be subject to change according to levels of understanding of students.