Probability and Measure
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WBMA02405 
Vaknaam  Probability and Measure 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester II b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Probability and Measure  
Leerdoelen  At the end of the course, the student is able to: 1) describe elementary notions as area and volume in terms of measures. 2) reproduce the elementary definitions and facts concerning measurable and integrable functions. 3) reproduce the main limiting theorems and to apply them in concrete situations. 4) apply the (nontrivial) transformation formulas involving the Jacobian determinant for computing integrals. 5) review several examples of standard function spaces of Lebesgue integrable functions. 6) outline the theory behind decomposition of measures. 7) express the main measuretheoretic concepts and results in the setting of Probability Theory. 8) define and relate the different modes of convergence of random variables in Probability Theory using measuretheoretic notions. 

Omschrijving  Measure and integration offers a general approach to the theory of integration based on measure theory. The starting point is an abstract framework through the study of collections of sets having desirable properties (the socalled sigmaalgebras), and of realvalued functions defined on these collections (the measures themselves). From this, one defines measurable functions, integrable functions and the Lebesgue integral. This is a far reaching extension of the well known theory of Riemann integration. The approach via measures provides a wide variety of applications: in particular about the interchange of various limiting procedures. Also, a connection with functional analysis is provided via the introduction of spaces of Lebesgue integrable functions. The course explores connections with probability theory and includes the mathematical concepts needed to understand stochastic processes.  
Uren per week  
Onderwijsvorm  Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(The final grade is determined on the basis of the grades of the 3 homework assignments and the written examination: the average of the three homework assignments counts for 40%, the written exam counts for 60%.) 

Vaksoort  bachelor  
Coördinator  prof. dr. J.P. Trapman  
Docent(en)  prof. dr. J.P. Trapman  
Verplichte literatuur 


Entreevoorwaarden  The course requires previous knowledge of: countable and uncountable sets, sequences and series of real numbers, related convergence notions, the analysis of realvalued functions defined on the real line; all these notions are acquired in the Analysis course. In the Analysis course, students are also expected to have acquired the ability of explaining and elaborating mathematical proofs. Skills in differentiating and integrating realvalued functions are also needed, and they are acquired in Calculus 1 and 2. Knowledge of elementary Probability notions, as in Probability Theory, is needed.  
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