## Foundations of probabilistic methods (lecture) (wykład) - 2019/2020

Course description
General information
 Lecturer: dr hab. August Zapała Organising unit: Faculty of Science and Health - Instytut Matematyki, Informatyki i Architektury Krajobrazu Number of hours (week/semester): 2/30 Language of instruction: English
Course objective
 General objectives of the course: Development of basic skills concerning mathematical description of random phenomena Detailed objectives: C1 - Studying mathematical methods used for the description of random phenomena C2 - Learning methods for calculating probabilities of random events, determining distributions of random variables and finding numerical parameters of probability distributions C3 - Learning about different modes of convergence of random variables C4 - Calculating the characteristic functions (Fourier transforms) C5 - Learning the basic limit theorems of probability theory
Prerequisites
 Prerequisities Mathematical analysis (elements of set theory, sequences and series of numbers, differential and integral calculus of functions of one and several variables)
Learning outcomes
 Learning outcomes for the subject KNOWLEDGE W1. Students give various definitions of probability and build mathematical models describing random phenomena and random experiments (K_W02) W2. Students list the most important discrete and continuous probability distributions (K_W02) W3. Students quote the basic theorems of probability theory (K_W02) SKILLS U1. Students use in practice various probability definitions, the law of total probability and the Bayes formula, examine the independence of random variables, calculate parameters of distributions for discrete and continuous random variables, calculate covariances and correlation coefficients, find equations of regression lines (K_U22) U2. Students recognize probability distributions based on characteristic functions (K_U22) U3. Students apply probabilistic methods for solving problems from various fields (K_U22) SOCIAL COMPETENCES (ATTITUDES) K1. Students formulate opinions on selected practical issues using tools of probability theory (K_K01)
Teaching method
 Teaching methods Lectures - conventional Classes - solving problems at the blackboard Assesment of classes - 2 written tests (4 problems at each test, each task 0-25 points, total max 4x25 points = 100 points), resit test To get a credit student should pass at least one of 2 tests, obtaining minimum 30 points, or pass a written examination. The final examination consists of two parts: written (50%) - verifying the ability to apply in practice the knowledge gained during lectures and classes, and oral (50%) - checking the theoretical knowledge acquired during the lecture. Evaluation criteria [0-30%) points - unsatisfactory (2) [30% -40%] - satisfactory (3) [40% -50%) - satisfactory plus (3.5) [50% -65%) - good (4) [65% -80%) - good plus (4.5) [80% -100%] - very good (5) Detailed rules of assessment are presented to students during the first class. Methods of checking acquired knowledge, skills and social competences: W1 - written and oral exam, tests, preparation for classes W2 - written and oral exam, tests, preparation for classes W3 - written and oral exam, tests, preparation for classes U1 - written and oral exam, tests, preparation for classes U2 - written and oral exam, tests, preparation for classes U3 - written and oral exam, tests, preparation for classes K1 - work and activity during classes Hours implemented as parts of the program study: Lectures 30 hours Classes 30 hours Total number of hours with the participation of an academic teacher 60 Number of ECTS credits with the participation of an academic teacher 3 Student\'s own work Preparation for classes - 20 hours Studying literature - 10 hours Preparing for tests and exam - 20 hours Total number of student\'s own work - 50 hours Number of ECTS credits 2 Total number of learning 60 + 50 = 110 hours Total number of ECTS credits for the module 3+2 = 5
Course content description
 The sample space, elementary events and random events. Fields and σ-fields of events. Classical and geometrical definitions of probability, examples of applications. Axioms of probability. Independence of events, fields and σ-fields of events. Conditional probability, the law of total probability and Bayes formula. Discrete probability spaces. Distribution function. Construction of a probability from distribution function on R. Multivariate distribution function and its connection with probability on a finite dimensional Euclidean space. Random variable, the law and distribution function of the random variable. Discrete and continuous distributions, probability density. Random vectors and multidimensional distributions. Marginal distributions of discrete and continuous random vectors. Independent random variables, criteria of independence for discrete and continuous random variables. Expectation and its properties. Variance, standard deviation, and their properties. Moments and central moments. Covariance and correlation coefficient, properties of the correlation coefficient. Lines of regression. Various modes of convergence of random variables (in distribution, in probability, almost surely and in mean). Markov’s and Chebyshev’s inequalities. Relationships between various modes of convergence. Complex random variables, independence and expectations of complex random variables. Characteristic functions and their properties. Lévy’s theorem (the inversion formula). Inversion formulas for discrete distributions. Inversion formulas for probability densities. The Lévy-Cramér theorem. The Lindeberg-Feller central limit theorem for a sequence of random variables, Lyapunov and Lindeberg-Lévy theorems (without proof). Weak law of large numbers – Khintchine\'s, Chebyshev\'s and Markov\'s theorems, classical criterion for convergence to a constant (without proof). Kolmogorov’s inequality, Kolmogorov’s criterion and the strong law of large numbers (without proof). Information concerning the simplest stochastic processes.
Forms of assessment
 Passing one of 2 tests, written and oral examination. GENERAL EVALUATION CRITERIA UNSATISFACTORY (W1) Student does not know the definition of probability, random variable, distribution and density (W2) Student does not know discrete and continuous probability distributions (W3) Student does not know characteristic functions and inversion formulas (W4) Student does not know limit theorems (U1) Student cannot build or analyze mathematical models of random experiments and calculate probabilities of random events (U2) Student cannot use the law of total probability and Bayes formula (U3) Student cannot give examples of discrete and continuous probability distributions and does not know practical applications of these distributions (U4) Student cannot determine parameters of discrete and continuous distributions, find equations of regression lines, calculate characteristic functions and use limit theorems and laws of large numbers (K1) Student does not formulate any precise questions to deepen his understanding of the subject SATISFACTORY (W1) Student knows the classical definition of probability and the definition of random variable (W2) Student knows the simplest discrete probability distributions (W3) Student knows some numerical characteristics of discrete random variables (W4) Student knows the selected limit theorems (U1) Student is able to build and analyze simple mathematical models of random experiments and calculate probabilities of random events in simple problems (U2) Student is able to apply the law of total probability (U3) Student is able to give simple examples of discrete probability distributions (U4) Student knows how to determine parameters of distribution for discrete random variables, find equations of regression lines, calculate characteristic functions (K1) Student formulates some questions in order to deepen his understanding of the topic GOOD (W1) Student knows various probability definitions, the definition of random variable and distribution function (W2) Student knows the most important discrete probability distributions (W3) Student knows characteristic functions and inversion formulas for discrete random variables (W4) Student knows the most important limit theorems (U1) Student is able to build and analyze mathematical models of random experiments using probability spaces and calculate probabilities of random events in classical problems (U2) Student is able to apply the law of total probability and Bayes formula in classical cases (U3) Student gives various examples of discrete and continuous probability distributions (U4) Student knows how to determine parameters of distributions for discrete random variables, find equations of regression lines, and calculate characteristic functions (K1) Student formulates various questions in order to deepen his understanding of the subject VERY GOOD (W1) Student knows various definitions of probability, random variable, joint and marginal probability distributions (W2) Student knows discrete and continuous probability distributions (W3) Student knows characteristic functions and inversion formulas for discrete and continuous random variables (W4) Student knows laws of large numbers and central limit theorems (U1) Student is able to build and analyze mathematical models of random experiments and calculate probabilities of random events in complex problems (U2) Student is able to apply the law of total probability and Bayes formula in non-standard cases (U3) Student is able to give various examples of discrete and continuous probability distributions and knows practical applications of these distributions (U4) Student knows how to calculate parameters of distributions for discrete and continuous random variables (K1) Student formulates precise questions in order to deepen his understanding of the topic