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 AD1M01MPE Mathematics for Economy Extent of teaching: 28+6 Guarantors: Roles: P,V Language ofteaching: CS Teachers: Completion: Z,ZK Responsible Department: 13101 Credits: 6 Semester: Z

Anotation:

Aim of this subject is to give the basic informations about probability, mathematical statistics and Markov chains and to show their applications, mainly in insurance mathematics. At the end of the course, bases of cluster analysis will be shown.

Course outlines:

 1 Random event, definition of probability. 2 Conditional probability, Bayes theorem. 3 Random variable, random vector - density, distribution function, expected value, variance; examples of discrete and continuous distributions. 4 Large numbers laws, central limit theorem. 5 Statistics - parameters estimations, testing of hypotheses. 6 Regression analysis. 7 Random processes - fundamental definitions. 8 Markov chains with discrete time - basic properties, random walk. 9 Markov chains with discrete time - transition matrix, Chapman-Kolmogorov equation, states classification. 10 Markov chains with continuous time - Wiener process, Poisson process. 11 General insurance - basic probability distributions of the number of events and claim amounts. 12 Technical reserves - indemnity reserve, triangular schemes, Markov chains in bonus systems. 13 Life insurance - premium in capital and annuity insurance. 14 Cluster analysis - basic definitions, methods of clustering.

Exercises outline:

 1 Probability of random event. 2 Conditional probability, Bayes theorem. 3 Distribution of random variable. 4 Discrete random variable - distribution function, expected value, variance. 5 Continuous random variable - density, distribution function, expected value, variance. 6 Central limit theorem. 7 Statistics - parameters estimations, testing of hypotheses. 8 Regression analysis. 9 Random processes - stationarity. 10 Markov chains with discrete and continuous time - transition matrix, classification of states, matrix of transition intensity. 11 Calculation of premium and reserves in general insurance. 12 Calculation of premium in capital insurance. 13 Calculation of premium in annuity insurance. 14 Basic methods of clustering.

Literature:

 [1] Papoulis, A.: Probability and Statistics, Prentice-Hall, 1990. [2] Stewart W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press 2009. [3] Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern actuarial risk theory. Kluwer Academic Publishers, 2004. [4] Gerber, H.U.: Life Insurance Mathematics. Springer-Verlag, New York-Berlin-Heidelberg, 1990. [5] Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. John Wiley & Sons, 2001.

Requirements:

Details are at http://math.feld.cvut.cz/helisova/mekA1M01MPE.html and http://math.feld.cvut.cz/helisova/mekAD1M01MPE.html respectively.

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