1. komponenta

Lecture type	Total
Lectures	45
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: In this course students will be introduced to the basic concepts and results of probability theory and statistics. The emphasis will be on discrete and continuous distributions.

COURSE DESCRIPTION AND SYLLABUS:
1-2) Basic concepts of probability. Sample space, events, probability as a ratio. Laplace model. Interpretations of probability (frequentist / a posteriori, subjective). Properties of probability, definition of a probability space (on an algebra of events and on a Sigma-algebra of events). Construction of a finite probability space, discussion of a countable probability space. Intuitive introduction to the concept of a distribution. Conditional probability, independence. Total probability formula, Bayes' formula.
3-4) Repetition of experiments. Product of discrete probability spaces, repetition of experiments, independence. Bernoulli scheme, binomial distribution, the concept of a binomial random variable. Normal approximation of the binomial distribution, the De Moivre-Laplace theorems (proof optional). Poisson approximation of a binomial random variable.
5-6) Discrete random variables. Definition of a random variable, distribution of a random variable, probability mass function, function of a random variable, random vector, joint probability function of a random vector, independence of random variables. Mathematical expectation, expectation of a sum, expectation of a function of a random variable, Markov's inequality. Variance, Chebyshev's inequality, the (weak) law of large numbers, the central limit theorem (without proof). Examples of discrete distributions - binomial, geometric, negative binomial, hypergeometric, Poisson.
7-8) Continuous distributions. Continuous random variable, probability density function, mathematical expectation and variance, comparison with the discrete random variable, examples (uniform, exponential, normal). Functions of a continuous random variable, change-of-variables formula. Distribution function of a random variable.
9-10) Continuous multidimensional distributions. Continuous random vectors, joint probability density function, independence of random variables. Distribution of functions of a random vector, sum, convolution, other operations, gamma distribution. Independent normal variables, Chi square - distribution, Student's t-distribution.
11-12) Foundations of statistics. Statistical data. Tabular and graphical presentation of a data set. Numerical characteristics of a data set (measures of central tendency, measures of variability).
Statistical dependence (contingency tables, correlation coefficient). Linear relationship between variables.
13-15) Population and sample. Population parameters and statistics. Elements of statistical inference. Parameter estimation. Confidence intervals. Statistical test, t-test, Chi square - test. Tests of homogeneity and independence of discrete variables (Chi square - test). Linear regression (estimation of the regression line, prediction).

1. Uvod u matematičku statistiku, Ž. Pauše, Školska knjiga, Zagreb, 1993.
2. Teorija vjerojatnosti, N. Sarapa, Školska knjiga, Zagreb, 2003.
3. Elements of Statistics, F. Daly, D. J. Hand, M. C. Jones, A. D. Lunn, K. J. McConway, Addison-Wesley, 1995.
4. Probability, J. Pitman, Springer Verlag, 1993.

Enrollment :
Attended : Mathematical analysis 2

Examination :
Passed : Mathematical analysis 2

Mandatory course - Regular study - Mathematics and Physics Education