Nonelementary Notes on Elementary Events

• Autorzy: Frič R.
• Konferencja: X Polish-Czech Mathematical School (10 ; 04-07.06.2003 ; Poraj near Częstochowa, Poland)
• Języki publikacji: EN

Abstrakty

EN

Our goal is to present simple examples illustrating the nature and role of elementary events and random variables in probability theory, both classical and operational (fuzzy). As stated in Płocki [10], in teaching probability we should concentrate on the construction of probability spaces and their properties, and not on the calculation of probability of various strange events (like hitting a bear if we can shoot three times, etc.). On a rather advanced level, Łoś [8] analyzed the constructions of probability spaces in the classical probability. J. Loś explained the nature and underscored the role of elementary events. Roughly, the events form a Boolean algebra, but some probability properties of the algebra depend on its representation via subsets and this is done via the choice of some fundamental subset of events and the choice of elementary events. Remember, choice! There are situations in which the classical probability model is not quite suitable (quantum physics, fuzzy models, c.f. Dvurečenskij and Pulmannová [3], Frič [5]), and I would like to present simple examples and simple models of such situations. In order to understand the generalizations, let me start with a well-known example of throwing two dice.

Słowa kluczowe

EN

probability theory; random variables

PL

teoria prawdopodobieństwa; zmienne losowe

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 2003
• Tom: Vol. 9
• Strony: 137--141
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Frič R.

• Catholic University in Ružomberok, Námestie A. Hlinku 56, 034 01 Ružomberok
• Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovak Republic

Bibliografia

• [1] S. Bugajski, Statistical maps I. Basic properties, Math. Slovaca 51, 321-342, 2001.
• [2] S. Bugajski, Statistical maps II. Operational random variables, Math. Slovaca 51, 343-361,2001.
• [3] A. Dvurečenskij, S. Pulmannová, New Trends in Quantum Structures, Kluwer Academic Publ., Dordrecht, Ister Science, Bratislava. 2000.
• [4] D. J. Foulis, M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24, 1331-1352, 1994.
• [5] R. Frič, Duality: random variables versus observables, Annales Academiae Pedagogicae Cracoviensis, Studia Mathematica (in press).
• [6] S. Gudder, Combinations of observables, Internat. J. Theoret. Phys. 31,695-704, 2000.
• [7] F. Kôpka, F. Chovanec, D-posets, Math. Slovaca 44, 21-34, 1994.
• [8] J. Łoś, Fields of events and their definition in the axiomatic treatment of probability, Studia Logica 9, 95-115 (Polish); 116-124 (Russian summary), 1960. 125-132 (English summary), 1960.
• [9] M. Papčo, On measurable spaces and measurable maps, (Preprint}.
• [10] A. Płocki, Pravděpodobnost kolem nás. Počet pravděpodobnosti v úlohách a problémech, Acta Universitatis Purkynianie 68, Studia Matematica IV, Ústí nad Labem, 2001.