Master
2019/2020

## Probability Theory and Mathematical Statistics

Type:
Compulsory course (Statistical Modelling and Actuarial Science)

Area of studies:
Economics

Delivered by:
Department of Statistics and Data Analysis

When:
1 year, 1, 2 module

Mode of studies:
offline

Master’s programme:
Statistical Modelling and Actuarial Science

Language:
English

ECTS credits:
5

### Course Syllabus

#### Abstract

This course aims to provide a solid introduction to probability theory and mathematical statistics. The fundamental concepts and mathematical tools for modeling and analysis of random phenomena will be presented and discussed.

#### Learning Objectives

- To provide students with the knowledge of the theoretical aspects and modeling tools related to probability theory.
- To provide students with the knowledge of elementary techniques to analyze probabilistic models.
- To present and study some fundamental distributions of random variables.
- To develop students' ability to apply the knowledge acquired during the course and to use probabilistic models in concrete situations, recognizing the appropriate frameworks and analytical tools related to the study.
- To outline the basic concepts and methods of mathematical and applied statistics.
- To give practical skills in applying statistical methods in applied research.
- To give an idea of the applied methods of multidimensional statistical analysis.

#### Expected Learning Outcomes

- The students are expected to learn the operations between sets, the notion of σ-algebra, Kolmogorov's axioms and properties of probability measures.
- The students are expected to learn the concept of random variable, some distribution functions, expected value and variance. Also, they will have the opportunity to relate random problems with random variables.
- The students will have the opportunity to model random experiments where the knowledge of continuous random variables is required. The concepts of expected value and variance for continuous random variables will be generalized. Also, the students will learn some of the main inequalities in Probability Theory.
- The concepts of random variables will be generalized to random vectors. It will be introduced the concepts of multivariate distributions.
- The students will learn about some convergences that are used in Probability Theory. LLN and CLT and their applications will be discussed.

#### Course Contents

- Introduction to Probability TheoryDiscrete probability. Sample space and operation on sets. Probability on a finite sample space. σ-algebra. Axioms of probability theory. Independent events. Conditional probability. Bayes theorem. Limit of events.
- Discrete random variablesGeneral definition of a random variable. Probability distribution of a discrete random variable. Cumulative Distribution Functions. Mean, variance and moments. Probability generating function. Independent random variables. Classical distributions: Bernoulli trials; Binomial distribution; Geometric distribution; Poisson distribution, .... Conditional expectation for discrete random variables.
- Continuous random variablesContinuous random variables. Elements of measure theory. Cumulative distribution function and probability density function. Distribution of a function of a real random variable. Moment, variance and Characteristic function. Classical distributions: Uniform; Normal; Exponential, Cauchy, ...; Markov, Chebyshev and Jensen Inequalities. Conditional density function and Conditional expectation.
- Random vectorsRandom vectors. Multivariate distributions. Joint laws, Marginal distribution. Covariance and correlation. Gaussian vectors and linear transformation. Chi-deux, Student and Fisher distributions, ...
- Convergences, LLN and CLTConvergence in probability. Limit of a sequence of random variables: convergence almost sure, in probability, in distribution. Convergence in Lp. The Slutsky theorem. Law of Large Numbers (LLN). The Central Limit Theorem (CLT) and its application.
- Statistical estimation of parameters. Samples. Property of estimators. Unbiasedness, efficiency, consistency
- Interval estimation. Standard confidence intervals for the parameters of a normal population. Confidence intervals for the mean, variance, difference of means, variances ratio, population proportion, the difference of proportions. The sample size
- Hypotheses testing. Type I, type II errors. P-value of the test. Tests on the values of the parameters of the normal population. Tests on mean, variance, differences, difference of means, variances ratio, population
- Estimation methods. The Method of Moments. Maximum Likelihood method. Their properties, examples. Information inequality (Fisher)
- Test statistics, Neumann-Pearson Lemma. Likelihood ratio test. Wald test. Lagrange multiplier test
- Goodness-of-fit tests. Contingency tables. Kolmogorov-Smirnov test
- Bayes approach to estimation
- One- and Two-way ANOVA
- Some concepts of non-parametric methods. Wilcoxon tests, run test. Rank correlation coefficients
- Sufficient statistic. Minimal sufficient statistic. Rao-Blackwell Theorem. Complete statistics. Lehmann-Sheffe Theorems

#### Assessment Elements

- Probability Theory: Quizzes and homework
- Probability Theory: Final exam of module 1
- Mathematical Satistics: quizzes and homework
- Mathematical Statistics: Exam

#### Interim Assessment

- Interim assessment (2 module)0.2 * Mathematical Satistics: quizzes and homework + 0.3 * Mathematical Statistics: Exam + 0.3 * Probability Theory: Final exam of module 1 + 0.2 * Probability Theory: Quizzes and homework

#### Bibliography

#### Recommended Core Bibliography

- Gut, A. (2005). Probability: A Graduate Course. New York, NY: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=155835
- Introduction to mathematical statistics, Hogg, R. V., McKean, J., 2014
- Ross, S. M. (2010). Introduction to Probability Models (Vol. 10th ed). Amsterdam: Elsevier Ltd. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=334597

#### Recommended Additional Bibliography

- A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, & M. I. Yadrenko. (2018). Probability Theory: Collection of Problems. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1790324
- Linde, W. (2017). Probability Theory : A First Course in Probability Theory and Statistics. [N.p.]: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1438416
- Stroock, D. W. (2011). Probability Theory : An Analytic View (Vol. 2nd ed). New York: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=357430
- Xia, X.-G. (2019). A Simple Introduction to Free Probability Theory and its Application to Random Matrices. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1902.10763