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# ch1pt1 - Chapter 1 Review of Probability Random Variable...

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Variable Concepts Review of Probability Concepts Dr. Lim HS Last Updated: 18 May 2009 c ± 2009 MMU

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Presentation outline Basics of Set Theory Fundamental Concepts in Probability Joint and Conditional Probability Independent Events Total Probability Theorem and Bayes’ Rule Combined Experiments c ± 2009 MMU Page 1/22
Basics of Set Theory Basic Deﬁnitions ƒ A set is speciﬁed by the content of two braces, e.g., A = { 6 , 7 , 8 , 9 } . ƒ a A : a is an element of set A . ƒ : empty set. ƒ A B : A is a subset of B . ƒ Two sets, A and C , are mutually exclusive if they have no common elements. ƒ The all encompassing set of objects under discussion is called the universal set , S . c ± 2009 MMU Page 2/22

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Basics of Set Theory Basic Deﬁnitions (cont.) ƒ Example 1. Consider the set of all positive integers smaller than 7: S = { x : 0 < x < 7 ,x an integer } S = { 1 , 2 , 3 , 4 , 5 , 6 } ƒ Example 2. Consider the set of all positive numbers smaller than 6: S = { x : 0 < x < 6 ,x a real number } c ± 2009 MMU Page 3/22
Basics of Set Theory Basic Set Operations ƒ The union of A and B is the set of all elements of A and B , denoted as A B . ƒ The intersection of A and B is the set of all elements common to A and B , denoted as A B . ƒ In particular, N [ n =1 A n = A 1 A 2 ∪ ··· ∪ A N N \ n =1 A n = A 1 A 2 ∩ ··· ∩ A N c ± 2009 MMU Page 4/22

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Basics of Set Theory Basic Set Operations (cont.) ƒ Complement of A is the set of all elements not in A , denoted as ¯ A = S - A . ƒ Algebra of sets: - Commutative law: A B = B A, A B = B A - Distributive law: A ( B C ) = ( A B ) ( A C ) A ( B C ) = ( A B ) ( A C ) - Associative law: ( A B ) C = A ( B C ) = A B C ( A B ) C = A ( B C ) = A B C - De Morgon’s law: ( A B ) = ¯ A ¯ B, ( A B ) = ¯ A ¯ B c ± 2009 MMU Page 5/22
Fundamental Concepts in Probability Deﬁnitions ƒ Random experiment: experiment whose result is unpredictable ƒ

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ch1pt1 - Chapter 1 Review of Probability Random Variable...

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