Subject: [HM] Classifications of dependencies of events in probability theory
From: Vorob'ov O.Yu. (vorob@scn.ru)
Date: Sat Jun 03 2000 - 00:15:48 EDT
Dear Colleagues,
thanks for interesting HM-discussions. I want to make my modest
contribution.
I think that the structure of dependencies between events is the key to
understand any aspects of the modern probability theory and the relationship
between probability and life. The Bernstein triplet {x,y,z} of events x,y
and z from algebra F of probability space (Omega, F, Prob) is well known:
B1:
xyz Prob
000 0
100 1/4
010 1/4
001 1/4
110 0
101 0
011 0
111 1/4
The Bernstein triplet illustrates a type of dependencies between pairwise
independent events which are dependent in common (S.N.Bernstein, Probability
Theory. 2-nd edition, Moscow, Leningrad: GTTI, in Russin, page 48). Now I
know only one more triplet of events with the same Bernstein type of
dependencies:
B2:
xyz Prob
000 1/4
100 0
010 0
001 0
110 1/4
101 1/4
011 1/4
111 0
Moreover now I know there are only seven types of dependencies of three
events including the Bernstein type, independent events and events with
arbitrary dependence. The following triplets illustrate 4 supplementary
types of dependencies which are different from Bernstein type, independent
events and from events with arbitrary dependence:
G1: G2: G3: G4:
xyz Prob xyz Prob xyz Prob xyz Prob
000 0 000 0 000 0 000 0
100 1/3 100 1/6 100 1/4 100 1/6
010 1/3 010 1/6 010 1/4 010 1/6
001 0 001 1/6 001 0 001 1/6
110 1/3 110 0 110 0 110 1/6
101 0 101 1/6 101 0 101 1/6
011 0 011 1/6 011 1/4 011 1/6
111 0 111 1/6 111 1/4 111 0
These examples of triplets with different types of dependencies of three
events were obtained by my postgraduate student Helen E. Goldenok in her
work to investigate a structure of dependencies of events.
I hope Colleagues can help me with several questions concerning
early classifications of dependencies of events.
I have been unable to find any information in sources available to me on
classifications of dependencies of events in probability theory. Can anyone
supply information and/or references?
Also I wish to ask whether you know about some scholars who investigated
dependencies of events in more details.
I thank you for your suggestions.
Best greetings from Siberia, Russia.
Oleg Vorob'ov
_________________________________________________
Oleg Yu. Vorob'ov voice: +7 3912 49-47-95
Institute of Computational Modeling
Russian Academy of Sciences FAX: +7 3912 43-98-30
Krasnoyarsk State University email: vorob@scn.ru
Akademgorodok 18-46
Krasnoyarsk, 660036, Russia
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