Mathematical Statistics, First Course

Aim

The course gives an introduction to the mathematical modelling of random experiments, with a special emphasis on applications in science, technology, and economics. After completing the course the student will be expected to be able to:

• M1 (Probabilities): identify experimental situations where random factors can affect the outcome, explain basic probability theoretical concepts and carry out probability calculations related to basic probability laws and theories.
• M2 (Stochastic variables): distinguish between different types of random variables, set up relevant probability-theoretic models for random experiments, and calculate important quantities in probability-theoretic models.
• M3 (Approximations): use the limit theories to analyze some random trials.
• M4 (Stochastic processes): set up and analyze probability-theoretic models for certain types of quantities that vary randomly in time, e.g. in the form of time-discrete Markov chains.

Content

• Random experiments and probabilities: Outcome spaces, events, combinatorics, probabilities, urn models, conditional probabilities, Bayes formula and the concept of independence.
• Random variables (one-dimensional): Discrete and continuous random variables, distribution function, probability function and density function.
• Measures of mode, dispersion, and correlation: Expected value, variance, and standard deviation.
• Probability distributions: Normal distribution, Binomial distribution, Poisson distribution, Exponential distribution, Uniform distribution, etc.
• Stochastic variables (Two-dimensional): Simultaneous/marginal probability function, simultaneous/marginal density function, and the concept of independence.
• Functions of random variables: Distribution derivation, distribution for sums of random variables and for maximum and minimum.
• Approximations: Law of large numbers, central limit value theorem and poisson approximation.
• Stochastic processes: Poisson process, discrete-time Markov chains.