SUBJECT
Title
Probability and statistics
Type of instruction
lecture + practical
Level
master
Faculty
Part of degree program
Credits
3+3
Recommended in
Semesters 1-4
Typically offered in
Autumn/Spring semester
Course description
- Probability space, random variables, distribution function, density function, expectation, variance, covariance, independence.
- Types of convergence: a.s., in probability, in Lp, weak. Uniform integrability.
- Characteristic function, central limit theorems.
- Conditional expectation, conditional probability, regular version of conditional distribution, conditional density function.
- Martingales, submartingales, limit theorem, regular martingales.
- Strong law of large numbers, series of independent random variables, the 3 series theorem.
- Statistical field, sufficiency, completeness.
- Fisher information. Informational inequality. Blackwell-Rao theorem. Point estimation: method of moments, maximum likelihood, Bayes estimators.
- Hypothesis testing, the likelihood ratio test, asymptotic properties.
- The multivariate normal distribution, ML estimation of the parameters
- Linear model, least squares estimator. Testing linear hypotheses in Gaussian linear models.
Readings
- J. Galambos: Advanced Probability Theory. Marcel Dekker, New York, 1995.
- E. L. Lehmann: Theory of Point Estimation. Wiley, New York, 1983.
- E. L. Lehmann: Testing Statistical Hypotheses, 2nd Ed., Wiley, New York, 1986.