Probability Theory and Mathematical Statistics
PREREQUISITES: Calculus linear algbra
OBJECTIVES
This course provides an excellent preparation for undergraduate students who are preparing for graduate study in statistics or statistically based areas such as econometrics, psychometrics, or biostatistics. It lays the essential mathematical basis for graduate-level courses in statistics or statistically based courses and many graduate programs from a variety of disciplines require or strongly recommend it for prospective. This courses also imparts to the student the important idea that real phenomena can be modeled stochastically using random variables/vectors and their distributions. These modeling aspects, which can be imparted through computer simulations, real experiments, and the use of historical data, should make the course very useful to students in the physical, engineering, biological and social sciences.
PROGRESSIVE ASSESSMENT
Coursework assignment (30%)
2 hour examination (70%)
RECOMMENDED TEXTS
1. Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press.
2. Jianya Liu(2007), Study Guide---- Probability theory and mathematical statistics, Shandong University Press.
3. Yintang Yuan (1990), Probability theory and mathematical statistics, Renmin University of China Press.
TOPICS TO BE COVERED IN THE LECTURES/ CLASS SCHEDULE
SECTION I: PROBABILITY THEORY
Week 1-2: random event and probability (Chapter 1 of “Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press” )
You will learn:
Discrete probability space.
The notion of the random experiment.
The random events and the operations over them.
The axiomatic introduction of the probabilities of the elementary events and the rules of computation of probability of any event.
The notion of the discrete probability space.
The theorems of addition and multiplication of the probabilities.
The conditional probability.
The independence of events.
The formula of the complete probability and the formula of Bayes.
Week 3-5 Random variable and its distribution(Chapter 2 of “Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press” )
Week 3::discrete random varible
Random variable and its main characteristics: Definition, examples and the main types of the random variables.
The possible values of the random variables.
Discrete random variable its probability distribution and main numerical characteristics, included: Binomial distribution, Geometric distribution, Poisson Distribution, 0-1 distribution, Paska distribution, etc
Week 4: Continuous random varible
Continuous random variable its probability distribution, density function and main numerical characteristics.
Uniform distribution
Normal (gaussian) law of probability distribution.
Exponential distribution
Week 5: distribution of function of random variable
Week 6-7 Two dimensional random variable and its distribution(Chapter 3 of “Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press” )
Week 6The notions of partial (marginal) and conditional distributions (on the example of two-dimensional discrete random variable). Independent random variables
Week7The notion of multidimensional law in a continuous case. The notions of partial (marginal) and conditional distributions (on the example of two-dimensional normal random variable).
Week 8-11 Numerical characteristic of random variable (Chapter 4 of “Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press” )
Week 8 mathematical expectation
Week 9 Variance
Week 10 Covariance and correlations
Week 11 Law of big numbers, and Central limit theorem
The special role of normal distribution: the central limit theorem.
The notion of asymptotic normality of sequence of random variables.
The formulation of the central limit theorem for independent identically distributed summands with finite variance (without proof);
de Moivre-Laplace theorem about asymptotic normality of binomial random variable (as a consequence of the central limit theorem);
SECTION 2: MATHEMATICAL STATISTICS
Week 12: The basis of statistics. (Chapter 4 of “Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press” )
You will learn
Population, sampling, and their basic characteristics: average, dispersion, asymmetry, excess, quantiles (percentage points), distribution and density functions;
Chi-square distribution;
t-distribution;
F-distribution;
Sampling distribution
Week 13-15 estimation of parameters, and fundamentals of Hypothesis testing (Chapter 5 of “Jianya Liu (2012), Probability theory and mathematical statistics, Higher Education Press” )
Week 13 Point estimate
Estimation by the method of moments
Maximum Likelihood estimation
their comparative analysis.
Week 14: Construction of interval estimates
Week 15: Basics of the statistical hypothesis testing theory.
Types of statistical criteria and their application: goodness of fit test (concerning the distribution function), homogeneity, series of observations stationarity, parametric criteria.
General scheme of any statistical criteria and its quality characteristics.
Week 16-17 Elements of regression and variance analysis.
Classical model of simple regression and classical method of least squares (OLS).
Statistical analysis of simple regression in framework of two-dimensional normal distribution.
Concept of variation analysis (one- and two-factor models)
Week 18: Examination