COURSE AIMS AND OBJECTIVES: In this course the students will be introduced to the basic notions and results of theory of probability and statistics. The accent will be given on discrete and continuous distributions.
COURSE DESCRIPTION AND SYLLABUS:
1. Basic notions in probability. Sample space, events, probabilty as a ratio. Laplace model. Interpretations of probability (frequentionist, i.e. posterior, subjective). Properties of probability, definition of probabiltiy space (algebra of events, and Sigma - algebra of events). Construction of finite probabiltiy space, discussion of a countable probability space. Introduction of distributions in an intuitive way. Conditional probability, independence. Bayeso formula.
2. Repeated trials. Product of discrete probabiliy spaces, repeated trials, independence. Bernoulli trials, binomial distributions, binomial random variables. Normal approximation of binomial distribution, Moivre - Laplace theorem. Poissono approximation of binomial random variable.
3. Discrete random variables. Definition of random variables, distributions of random variables, probability density function, functions of random variables, random vectors, probability density function of random vectors, independence of random variables. Mathematical expectation, expectation of a sum, expectation of a function of random variable, Markov inequality. Variance, Chebishev inequality, (weak) law of large numbers, central limit theorem (without proof). Examples of discrete distributions - binomial, geometric, negative binomial, hipergeometric, Poisson.
4. Continuous distributions. Continuous random variable, probability density function, mathematical expectation and variance, comparison with discrete random varaibles, examples (uniform, exponential, normal). Functions of continuous random variables. Functions of distributions of random variables.
5. Continuous multidimensional distributions. Continuous random vectors, probability density function, independence of random variables. Distribution function of random vectors, sum, convolution, other operations, gamma distribution. Independent normal variable, Chi square - distribution, Student t - distribution.
6. Elements of statistics. Statistical data. Tables and graphs. Numerical characteristics of statistical data (mean, measures of variability). Statistical dependencies (contingency tables, coefficient of correlation). Linear dependency between variables. Population and sample. Population parameters and statistics. Elements of statistical inference. Parameter estimation. Confidence intervals. Statistical tests, t - test, Chi square - test. Testing distribution homogeneity and independency in contingency tables (Chi square - test). Linear regression (estimation of simple linear model, prediction).