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Probability theory I*

General data

Course ID:	1000-114bRP1*
Erasmus code / ISCED:	11.1 The subject classification code consists of three to five digits, where the first three represent the classification of the discipline according to the Discipline code list applicable to the Socrates/Erasmus program, the fourth (usually 0) - possible further specification of discipline information, the fifth - the degree of subject determined based on the year of study for which the subject is intended. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Probability theory I*
Name in Polish:	Rachunek prawdopodobieństwa I*
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Obligatory courses for 2nd grade JSEM Obligatory courses for 2nd grade JSIM (3I+4M) Obligatory courses for 2nd grade JSIM (3M+4I) Obligatory courses for 2rd grade Mathematics Obligatory courses for 3rd grade JSIM (3I+4M)
ECTS credit allocation (and other scores):	7.50 Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	Polish
Type of course:	obligatory courses
Prerequisites (description):	(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusie przedmiotu Analiza matematyczna II.1.
Short description:	Kolmogorov axioms. Basic probabilities. Random variables, probability distributions, and their parameters. Independence. Convergence of random variables. Basic limit theorems: Poisson theorem, weak and strong laws of large numbers, de Moivre-Laplace theorem.
Full description:	Kolmogorov axioms. Properties of probability measures. Borel-Cantelli lemma. Conditional probability. Bayes' theorem.. Basic probabilities: classical probability, discrete probability, geometric probability. Random variables (one- and multidimensional), their distributions. Distribution functions. Discrete and continuous distributions. Distribution densities. Parameters of distributions: mean value, variance, covariance. Chebyshev inequality. Independence of: events, sigma-algebras, random variables. Bernoulli (binomial) process. Poisson theorem. Distrubution of sums of independent random variables. Convergence of random variables. Laws of large numbers: weak and strong. De Moivre-Laplace theorem. The program is in principle the same as for the basic lecture. However, topics will be treated more deeply and often in a more general way. The lecture is addressed to students with deeper interest in the subject, eager to tackle related exercises and problems.
Learning outcomes:	(in Polish) Student 1. Zna definicję przestrzeni probabilistycznej i podstawowe własności prawdopodobieństwa. 2. Zna podstawowe schematy probabilistyczne: prawdopodobieństwo "klasyczne", dyskretne, geometryczne. Potrafi operować przykładami. 3. Zna lemat Borela-Cantellego. 4. Zna wzór na prawdopodobieństwo całkowite i wzór Bayesa. 5. Zna pojęcie zmiennej losowej i jej rozkładu. Potrafi podać najważniejsze rozkłady dyskretne i ciągłe. Zna pojęcie dystrybuanty oraz jej własności. Potrafi znajdować rozkłady zmiennych losowych będących funkcjami innych zmiennych losowych o znanych rozkładach. 6. Zna pojęcia wartości oczekiwanej, wariancji i kowariancji oraz potrafi obliczać te wielkości. Zna nierówność Czebyszewa. 7. Zna pojęcia niezależności zdarzeń i sigma-ciał oraz niezależności zmiennych losowych. Umie znaleźć rozkład sumy niezależnych zmiennych losowych. Zna schemat Bernoulliego i twierdzenie Poissona. 8. Potrafi rozstrzygać o zbieżności ciągów zmiennych losowych. Zna relacje między różnymi rodzajami zbieżności (prawie na pewno, według prawdopodobieństwa, w L^p) i potrafi je zilustrować przykładami. 9. Zna słabe i mocne prawo wielkich liczb. 10. Zna Centralne Twierdzenie Graniczne w postaci de Moivre'a-Laplace'a.
Assessment methods and assessment criteria:	(in Polish) Przedmiot kończy się egzaminem.

Classes in period "Summer semester 2024/25" (past)

Time span:	2025-02-17 - 2025-06-08	Selected timetable range: weekly course term Go to timetable MO WYK CW TU W CW TH FR
Type of class:	Classes, 45 hours more information Lecture, 30 hours more information
Coordinators:	Witold Bednorz
Group instructors:	Witold Bednorz
Students list:	(inaccessible to you)
Credit:	Course - Examination Lecture - Examination

Classes in period "Summer semester 2025/26" (future)

Time span:	2026-02-16 - 2026-06-07	Selected timetable range: weekly course term Go to timetable
Type of class:	Classes, 45 hours more information Lecture, 30 hours more information
Coordinators:	Rafał Latała
Group instructors:	Rafał Latała
Students list:	(inaccessible to you)
Credit:	Course - Examination Lecture - Examination

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