Foundations of the theory of probability

§5. Independence 9 Bernsteinisactuallydedicatedtothefundamental investigation of series of independent random variables. Though ...

10 I. ElementaryTheoryofProbability Amongtherequationsin (2), thereareonlyr-r 1 -r 2 .. . -r n + n 1 independent equations ...

§ Independence 11 1 = 1) fortheindependenceoftwoeventsA x andA 2 : ?UiA 2 ) =P(A 1 )P(A 2 ). (5) Thesystemofequations (2) redu ...

12 I. ElementaryTheoryofProbability five probabilities P(A^ } ) is that the conditional probability of the resultsA q w of exper ...

§ ConditionalProbabilitiesasRandomVariables, MarkovChains 13 P m (A?)= P(Af) q=\,2,...,r 2 . Givenanydecompositions (experimen ...

Chapter II INFINITE PROBABILITY FIELDS § AxiomofContinuity Wedenote by 2)A m ,asis customary, theproductofthe sets m A m (whet ...

§1. AxiomofContinuity 15 n limP(^ B) =P(o)= 0. Allexamplesof finite fieldsofprobability, inthefirstchapter, satisfy,therefore, A ...

16 II. InfiniteProbabilityFields anyfield g.*Wecan, therefore, definetheconcept ofafield of probabilityinthefollowingway:LetEbea ...

§ BorelFieldsofProbability 17 ExtensionTheorem: Itisalways possible to extend a non- negative completely additive set function ...

18 II. InfiniteProbabilityFields actualrandomevents,the extendedfield ofprobability (B%, P) willstill remainmerelyamathematical ...

§ 3.ExamplesofInfiniteFieldsofProbability 19 P(A) on 3 which satisfies thecondition P(E) = As is well known 3 ,suchafunctionis ...

20 II. InfiniteProbabilityFields The investigation offields of probability oftheabovetype issufficientforallclassicalproblems in ...

ChapterIII RANDOM VARIABLES § Probability Functions GivenamappingofthesetEintoasetE'consistingofany- typeofelements,i.e.,asing ...

22 III.RandomVariables LetA'andB'betwosetsof $ (M >.Theirpre-imagesAandB belongthento J. Since % isafield,thesetsAB,A + B, an ...

§ DefinitionofRandomVariablesandofDistributionFunctions 23 Since g ( *> isafield, thenalongwiththeintervals (-oo,«a) it con ...

24 III. Random Variables F (x) (a) (cf. §3, IIIinChap. II).Sinceourmaininterestliesin thesevaluesof P (x) (A), thedistributionfu ...

§3. Multi-dimensionalDistributionFunctions 25 subsets of spaceR n anda probability function pfe»»*»•••»•*> (4') definedon gf' ...

26 III. RandomVariables limF(« lf a 2 , ...,a n ) =F(a v .. .,«,_ lf -oo,a i+1 , .. .,a n ) = 0, (7) limyfo,a,, .. .,a n ) = F(+ ...

§ ProbabilitiesinInfinite-dimensionalSpaces 27 p<*.,*a ,...,**>(,4)=pttk.*.....-^^-!^)}. (ii) For thecorresponding distr ...

28 III. RandomVariables /(**.**.-••»**,)=-<). (1) Inordertodetermine an arbitrary cylinderset P Ml ^ ... ^ (A') by sucharelat ...