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# lec.02 - ENGG2430C Lecture 2 Probability Model and Axioms...

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ENGG2430C Lecture 2: Probability Model and Axioms Minghua Chen ([email protected]) Information Engineering The Chinese University of Hong Kong Reading: Ch. 1.1 1.2, and 1.3 of the textbook M. Chen ENGG2430C lecture 2 1 / 23

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Review: Set and Set Operations A set is a collection of objects: S = { x 1 , x 2 ,..., x n } The whole space Ω , and the empty set /0 S c = { x Ω | x / S } S T = { x | x S or x T } S T = { x | x S and x T } M. Chen ENGG2430C lecture 2 2 / 23
Review: Venn Diagram (from the textbook) M. Chen ENGG2430C lecture 2 3 / 23

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Exercise Let A = {set of all persons taller than 1.6 meters}, B = {set of all persons heavier than 60kg}, C = {set of all persons who wear glasses}. Can you relate: (Draw Venn diagrams and also express your mathematics statement in terms of plain English) A B and A B ? ( A B ) c and A c B c ? Answer: ( A B ) c = A c B c If A B , what can you say about the relationship between A c and B c ? What is the corresponding statement in English? I Answer: A c B c . M. Chen ENGG2430C lecture 2 4 / 23
Venn Diagrams of De Morgan’s Laws when n=2 ( A S B ) c A c T B c M. Chen ENGG2430C lecture 2 5 / 23

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De Morgan’s Laws Two particularly useful properties are given by De Morgan’s laws which state that I ( n S n ) c = T n S c n , I ( T n S n ) c = n S c n Proof of ( n S n ) c T n S c n I Suppose that x ( n S n ) c .Then, x / ∈ ∪ n S n , which implies that for every n , we have x / S n . I Thus, x belongs to the complement of every S n , and x T n S c n . M. Chen ENGG2430C lecture 2 6 / 23
Probability Model What is probability model? I A mathematical description of an uncertain situation. Elements of a probability model I The sample space Ω I The probability law (D. P. Bertsekas & J. N. Tsitsiklis, Introduction to Probability, Athena Scientific Publishers, 2002) M. Chen ENGG2430C lecture 2 7 / 23

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Probability Model: Sample Space A list of all possible outcomes The list must be: I Mutually exclusive
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