Course title:
Probability theory
Primary programme:
Fizikus mérnök BSc
ECTS credits:
5
Course type:
compulsory
Number of lectures per week:
2
Number of practices per week:
2
Number of laboratory exercises per week:
0
Further knowledge transfer methods:
-
Grading:
Examination
Special grading methods:
-
Semester:
3
Prerequisites:
Multivariable calculus
Responsible lecturer:
Dr. Péter Bálint, associate professor, PhD
Lecturers and instructors:
Course description:
1. Introduction: empirical background, sample space, events, probability as a set function. Enumeration problems, inclusion-exclusion formula, urn models, problems of geometric origin.
2. Conditional probability: basic concepts, multiplication rule, law of total probability, Bayes formula, applications. Independence.
3. Discrete random variables: probability mass function, Bernoulli, geometric, binomial, hypergeometric and negative binomial distributions. Poisson approximation of the binomial distribution, Poisson distribution, Poisson process, applications.
4. General theory of random variables: (cumulative) distribution function and its properties, singular continuous distributions, absolutely continuous distributions and probability density functions. Important continuous distributions: uniform, exponential, normal (Gauss), Cauchy. Distribution of a function of a random variable, transformation of probability densities.
5. Quantities associated to distributions: expected value, moments, median, variance and their properties. Computation for the important distributions. Steiner formula. Applications.
6. Joint distributions: joint distibution, mass and density functions, marginal and conditional distributions. Important joint distributions: polynomial, uniform and mutlidimensional normal distribution. Sums of independent variables, convolution, Conditional distribution and density functions. Conditional expectation and prediction, tower rule, conditional variance. Vector of expected values, Covariance matrix, Cauchy-Schwartz inequality, correlation. Indicator random variables.
7. Weak Law of Large Numbers: Bernoulli’s Law of Large Numbers, Markov and Chebyshev inequality. Weak Law of Large numbers in full generality.
8. Normal approximation of binomial distribution: Stirling formula, de Moivre-Laplace theorem. Applications. Normal fluctuations. Central Limit Theorem.
Reading materials:
William Feller: An Introduction to Probability Theory and its Applications, 3rd Edition, Wiley, ISBN-13: 978-8126518050
Sheldon Ross: A First Course in Probability, 10th Edition, Pearson, ISBN-13: 978-1292269207
List of competences:
Please find the detailed list, as quoted from the Hungarian training and outcome requirements of the Physicist Engineer program, in the Hungarian version of the course description.