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# ch2 - Chapter 2 Basic Concepts of Probability Theory...

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Chapter 2 Basic Concepts of Probability Theory ENCS6161 - Probability and Stochastic Processes Concordia University

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Specifying Random Experiments Examples of random experiments: tossing a coin, rolling a dice, the lifetime of a harddisk. Sample space: the set of all possible outcomes of a random experiment. Sample point: an element of the sample space S Examples: S = { H,T } S = { 1 , 2 , 3 , 4 , 5 , 6 } S = { t | 1 < t < 10 } Event: a subset of a sample space A S A = { H } A = { 2 , 4 , 6 } A = ENCS6161 – p.1/16
The Axioms of Probability A probability measure is a set function P ( · ) that satisfies the following axioms. 1. P ( A ) 0 2. P ( S ) = 1 3. If A B = , then P ( A B ) = P ( A ) + P ( B ) 4. If A 1 ,A 2 , · · · are events s.t A i A j = for all i negationslash = j then P bracketleftbig k =1 A k bracketrightbig = k =1 P ( A k ) Corollary 1: P ( A c ) = 1 - P ( A ) Corollary 2: P ( A ) 1 A Corollary 3: P ( ) = 0 ENCS6161 – p.2/16

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The Axioms of Probability Corollary 4: If A 1 · · · A n are mutually exclusive, i.e. A i A j = , i negationslash = j then P ( n k =1 A k ) = n k =1 P ( A k ) Corollary 5: P ( A B ) = P ( A ) + P ( B ) - P ( A B ) A B A B Union bound: P ( A B ) P ( A ) + P ( B ) ENCS6161 – p.3/16
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ch2 - Chapter 2 Basic Concepts of Probability Theory...

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