Foundations of the theory of probability

§ 1.ConditionalProbabilities 49 0(A) = //(f) P(dE). A Inorderto applythistheoremtoourcase,weneedtoprove 1° that Q(A) = P(Bu-HA)) ...

50 V. ConditionalProbabilitiesandMathematicalExpectations allowustocarryovermanyotherbasicpropertiesofthe absolute probabilityP( ...

§ ConditionalProbabilities withRespecttoaKandomVariable 51 oppositepoints forourpoles, so thateachmeridiancirclewillbe uniquel ...

52 V. ConditionalProbabilitiesandMathematicalExpectations P x (a;B) was defined in § 1 except on a set G, which is suchthatP (*& ...

§ ConditionalMathematicalExpectations 53 E{u*A}(y) =E {uc^ } E tt (y) f (1) iscalled(if it exists)theconditionalmathematical e ...

54 V. ConditionalProbabilitiesandMathematicalExpectations Ifuandvaretwofunctionsoftheelementaryevent £, then thecouple (u,v) can ...

§ ConditionalMathematicalExpectations 55 isthenecessary conditionfortheexistenceofE(y) (seeChap.IV, § 1). Fromthis convergence ...

56 V. ConditionalProbabilitiesandMathematicalExpectations inthesenseof §1, Chap. IV, so that (12) is onlya symbolic expression. ...

ChapterVI INDEPENDENCE; THE LAW OF LARGE NUMBERS § Independence Definition 1 :Twofunctions,uandvof |, aremutuallyinde- pendent ...

58 VI.Independence;TheLawofLarge Numbers (2). Conversely sinceP v (uczA) isuniquelydeterminedby (4) towithinprobability zero,the ...

§ 2.IndependentRandomVariables 59 thespaceR n ,then condition (1) is also sufficientforthemutual independence of the variables x ...

60 VI. Independence;TheLawofLargeNumbers InordertoshowthisitisenoughtotakeR M forthebasicsetE andB% M forthefield g, andtodefine ...

§ 3.TheLawofLargeNumbers 61 wherethex lf x 2 , ... , x n areuncorrelatedinpairs,wecaneasily computethat o 2 (s)=o 2 (*,) + o*(x ...

62 VI. Independence;TheLawofLargeNumbers Ifs n -E(s n ) are uniformly bounded: \s n -E(s n) \^M, thenfromtheinequality (9) in § ...

§ 3.The Law of LargeNumbers 63 We set Xnl;=%k if I ^fc-mfc | ^ n, Xnk ~~ otherwise, c* _ Xn +*„*+ ••• +*„ « jrelations k=n ZP{|* ...

64 VI. Independence;TheLawofLargeNumbers Then s n — E(s„)=z x + z 2 + 4- z n, E{z nk) =EE 3ll 9( 8 ... 9lifc (s n) — E ...

§ NotesontheConceptofMathematical Expectation 65 asageneralizedmathematical expectation.Welosein thiscase, ofcourse,severalsim ...

66 VI. Independence;TheLawofLargeNumbers (Chap. IV, §2) stillhold;ingeneral,however,theexistenceof E*|x | doesnotfollowfromtheex ...

§ 5.StrongLawofLargeNumbers;ConvergenceofSeries 67 where the variables x n are mutuallyindependent. A sufficient 8 conditionfort ...

68 VI. Independence;TheLawofLargeNumbers Theninorderthatseries ( 1 ) convergewiththeprobabilityone, itisnecessaryandsufficient 1 ...