elemprob-page4

# elemprob-page4 - Typically we will take F to be all subsets...

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Typically we will take F to be all subsets of S , and so (i)-(iii) are automatically satisﬁed. The only times we won’t have F be all subsets is for technical reasons or when we talk about conditional expectation. So now we have a space S , a σ -ﬁeld F , and we need to talk about what a probability is. There are three axioms: (1) 0 P ( E ) 1 for all events E . (2) P ( S ) = 1 . (3) If the E i are pairwise disjoint, P ( i =1 E i ) = i =1 P ( E i ) . Pairwise disjoint means that E i E j = unless i = j . Note that probabilities are probabilities of subsets of S , not of points of S . However it is common to write P ( x ) for P ( { x } ). Intuitively, the probability of E should be the number of times E occurs in n times, taking a limit as n tends to inﬁnity. This is hard to use. It is better to start with these axioms, and then to prove that the probability of E is the limit as we hoped. There are some easy consequences of the axioms.
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## This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.

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