[RP]Lecture Note I - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) I-1 C1002900 RP Lecture Handout I: Axioms of Probability Theory Reading: Chapters 1, 2 Random processes are widely used to model systems in engineering and scientific ap- plications. We adopt the most widely used framework of probability and random processes, namely: the one based on Kolmogorov’s axioms of probability. 1.1 Probability Space A probability space is a triplet  ,, P . The first component is a nonempty set. Each element of is called an outcome and is called the sample space . The second com- ponent is a set of subsets of called events. The set of events is assumed to be a -algebra (or -field ), meaning it satisfies the following axioms: A.1  . A.2 If A then c A . A.3 If , A B then AB  . If . . . AA 12 is a sequence of elements in then ii A 1 . If is a -algebra and , A B , then AB by A.2, A.3 and the fact that c cc A BAB  . By the same reasoning, if . . . is a sequence of elements in a -algebra , then i i A 1 . Events i A , iI , indexed by a set I , are called mutually exclusive (or disjoint ) if the in- tersection ij  for all , ij I with . The final component P of the triplet P is a probability measure on satisfying the following axioms:
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) I-2 P.1  PA 0 for all A . P.2 If A and B are mutually exclusive, then ( . finite additivity) PA B PA PB  If ,, . . . AA 12 is a sequence of mutually exclusive events in then (coun . table or infinite additivity) ii P A      11 P.3 P  1. We establish a continuity property of probability measures which is analogous to con- tinuity of functions on n . If . . . is a sequence of events such that  A 3 , then we can think that j A converges to the set A 1 as j  . The follow- ing theorem states that in this case, j P A converges to the probability of the limit set as j . Theorem 1.1 (Continuity of probability): Suppose . . . is a sequence of events. (a) If (increasing family of sets), then lim jj P AP A 1 . (b) If  (decreasing family of sets), then lim P A 1 . Proof: Suppose . Let D A and iii D  1 for i 2. Then   j j i i P D 1 for each j 1 . Therefore
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) I-3     a lim lim j j ii jj i i i i P AP D P D PD PA       11 1 1 where (a) follows from Axiom P.2. This proves Theorem 1.1 (a). Theorem 1.1 (b) can be
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[RP]Lecture Note I - Kyung Hee University Department of...

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