Difference between revisions of "Probability measure"
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A real non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749301.png" /> on a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749302.png" /> of subsets (events) of a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749303.png" /> (the space of elementary events) forming a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749305.png" />-field (i.e. a set closed with respect to countable set-theoretic operations) such that | A real non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749301.png" /> on a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749302.png" /> of subsets (events) of a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749303.png" /> (the space of elementary events) forming a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749305.png" />-field (i.e. a set closed with respect to countable set-theoretic operations) such that |
Revision as of 19:05, 2 April 2012
probability distribution, probability
2020 Mathematics Subject Classification: Primary: 60-01 [MSN][ZBL]
A real non-negative function on a class
of subsets (events) of a non-empty set
(the space of elementary events) forming a
-field (i.e. a set closed with respect to countable set-theoretic operations) such that
![]() |
if for
(
-additivity).
Examples of probability measures.
1) ;
is the class of all subsets of
;
(this probability measure corresponds to a random experiment consisting in throwing a symmetrical coin; if heads correspond to 1 while tails correspond to 2, the probability of throwing heads (tails) is 1/2);
2) ;
is the class of all subsets of
;
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where (the Poisson distribution);
3) ;
is the class of Borel subsets of
;
![]() |
(the normal distribution);
4) is the space of continuous real functions
on
that vanish at the point zero;
is the class of Borel subsets with respect to the topology of uniform convergence;
is the measure which is uniquely defined by the formula
![]() |
![]() |
![]() |
where is an arbitrary natural number and
(the Wiener measure).
References
[1] | A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) MR0032961 |
[2] | B.V. Gnedenko, "The theory of probability" , Chelsea, reprint (1962) (Translated from Russian) MR0149513 Zbl 0102.34402 |
Comments
References
[a1] | P. Billingsley, "Probability and measure" , Wiley (1979) MR0534323 Zbl 0411.60001 |
Probability measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_measure&oldid=23647