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, inclusionexclusion 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, CauchySchwartz 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 MoivreLaplace theorem. Applications. Normal fluctuations. Central Limit Theorem.
Reading materials:
William Feller: An Introduction to Probability Theory and its Applications, 3rd Edition, Wiley, ISBN13: 9788126518050
Sheldon Ross: A First Course in Probability, 10th Edition, Pearson, ISBN13: 9781292269207
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.