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[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 A B . If , ,... A A 1 2 is a sequence of elements in then i i A 1 . If is a -algebra and , A B , then AB by A.2, A.3 and the fact that c c c AB A B . By the same reasoning, if , ,... A A 1 2 is a sequence of elements in a -algebra , then i i A 1 . Events i A , i I , indexed by a set I , are called mutually exclusive (or disjoint ) if the in- tersection i j A A   for all , i j I with i j . 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 P A 0 for all A . P.2 If A and B are mutually exclusive, then ( . finite additivity) P A B P A P B If , ,... A A 1 2 is a sequence of mutually exclusive events in then (coun . table or infinite additivity) i i i i P A P A 1 1 P.3 P   1 . We establish a continuity property of probability measures which is analogous to con- tinuity of functions on n . If , ,... A A 1 2 is a sequence of events such that A A 1 2 A 3 , then we can think that j A converges to the set i i 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 , ,... A A 1 2 is a sequence of events. (a) If A A 1 2 (increasing family of sets), then lim j j i i P A P A  1 . (b) If A A 1 2 (decreasing family of sets), then lim j j i i P A P A  1 .
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  • Spring '10
  • HyungdongShin
  • Probability theory, Probability space, KYUNG HEE UNIVERSITY, Hee University Department, Kyung Hee University Department of Electronics, Prof. Hyundong Shin

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