STATSLIDE2 - AXIOMATIC PROBABILITY AND POINT SETS The...

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Unformatted text preview: AXIOMATIC PROBABILITY AND POINT SETS The axioms of Kolmogorov. Let S denote an event set with a probability measure P defined over it, such that probability of any event A ⊂ S is given by P ( A ). Then, the probability measure obeys the following axioms: (1) P ( A ) ≥ , (2) P ( S ) = 1 , (3) If { A 1 ,A 2 ,...A j ,... } is a sequence of mutually exclusive events such that A i ∩ A j = ∅ for all i,j , then P ( A 1 ∪ A 2 ∪···∪ A j ∪··· ) = P ( A 1 ) + P ( A 2 ) + · · · + P ( A j ) + · · · . The axioms are supplemented by two definitions: (4) The conditional probability of A given B is defined by P ( A | B ) = P ( A ∩ B ) P ( B ) , (5) The events A,B are said to be statistically independent if P ( A ∩ B ) = P ( A ) P ( B ) . This set of axioms was provided by Kolmogorov in 1936. 1 The rules of Boolean Algebra. The binary operations of union ∪ and intersection ∩ are roughly analogous, respectively, to the arithmetic operations of addition + and multiplication × , and they obey a similar set of laws which have the status of axioms: Commutative law: A ∪ B = B ∪ A , A ∩ B = B ∩ A , Associative law: ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ), ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ), Distributive law: A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ),...
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This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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STATSLIDE2 - AXIOMATIC PROBABILITY AND POINT SETS The...

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