probability distribution, probability
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 ;
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).
|[K]||A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) MR0032961|
|[G]||B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)|
|[B]||P. Billingsley, "Probability and measure" , Wiley (1979) MR0534323 Zbl 0411.60001|
Probability measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Probability_measure&oldid=26924