HISTORIA MATEMATICA Saturday, June 3 2000 Volume 02 : Number 063
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
[HM] Classifications of dependencies of events in probability theory
Re: [HM] Mathematics as Theater
Re: [HM] Mathematics as Theater
Re: [HM] Mathematics in Literature
Please see the end of this digest.
----------------------------------------------------------------------
Date: Sat, 3 Jun 2000 11:15:48 +0700
From: "Vorob'ov O.Yu."
Subject: [HM] Classifications of dependencies of events in probability theory
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
------------------------------
End of HISTORIA MATEMATICA V2 #63
*********************************
When replying to a message, PLEASE delete all parts of the digest to which
you are not alluding! This practice will save downloading time, and disc
space. Moreover, PLEASE maintain the SUBJECT LINE of the individual post
that you are replying, unless there is good reason to change it.
Please note that if you wish to reply just to the author of a posting, you
must copy his/her email address out of that individual message.
HISTORIA MATEMATICA Thursday, June 8 2000 Volume 02 : Number 068
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
[HM] Cayley
Re: [HM] The Universal History of Numbers
Re: [HM] The Universal History of Numbers
[HM] on dependencies of events in probability theory
[HM] Commentary on Euclid
Re: [HM] The Universal History of Numbers
Re: [HM] The Universal History of Numbers
Re: [HM] The Universal History of Numbers
[HM] Ludolph van Ceulen
Please see the end of this digest.
----------------------------------------------------------------------
Date: Wed, 7 Jun 2000 13:58:23 GMT0BST
From: "Dr J Stoyanov"
Subject: [HM] on dependencies of events in probability theory
Dear Colleagues,
The topic "independence/dependence" of random events and random variables
was and is among my favourite ones during the last 15 years.
I have proposed a new idea how to analyze the independence/dependence
properties of a collection of random elements and have established some
striking results in the area.
Also I proposed a "Dependency Measure", i.e. an easy way to calculate
how much is the dependency in a given collection of random elements. It
is a single number between 0 and 1... Many people found this interesting
and useful...
You may look at my book:
Stoyanov, J: "Counterexamples in Probability", 2nd ed, John Wiley &
Sons, 1997
or at my papers (sorry not on www...):
in "The American Statistician", vol 47 (1993), pp. 112-115
in "Statist & Probab Letters", vol 23 (1995), pp. 108-115
and in the book "Stochastic Processes and Related Topics" (In Memory
of S Cambanis). Eds Karatzas, I. et all Birkhauser, Boston, 1998,
pp. 257-275.
I have also something more which is interesting but still not published.
There are, of course, related works by others.
I would like to know about similar work of colleagues.
Best regards: Jordan
Dr Jordan Stoyanov
School of Mathematics & Statistics
University of Newcatsle
Newcastle upon Tyne NE1 7RU
UNITED KINGDOM
e-mail: jordan.stoyanov@ncl.ac.uk (l="el")
fax: +44 191 222 8020
------------------------------
End of HISTORIA MATEMATICA V2 #68
*********************************
When replying to a message, PLEASE delete all parts of the digest to which
you are not alluding! This practice will save downloading time, and disc
space. Moreover, PLEASE maintain the SUBJECT LINE of the individual post
that you are replying, unless there is good reason to change it.
Please note that if you wish to reply just to the author of a posting, you
must copy his/her email address out of that individual message.