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Example_class_2_slides - Example class 2 STAT1301...

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Example class 2 STAT1301 Probability and Statistics I Chan Chi Ho, Chan Tsz Hin & Shi Yun September 22, 2011 Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2
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Review: Set Theory and Mathematical Theory of Probability De Morgan’s law n [ i = 1 E i ! c = n \ i = 1 E c i , n \ i = 1 E i ! c = n [ i = 1 E c i , where n can also be . Language of Probability 1 Mutually exclusive A 1 , A 2 , ··· , A n are mutually exclusive if A i A j = φ for all i 6 = j . 2 Exhaustive A 1 , A 2 , ··· , A n are exhaustive if A 1 A 2 ∪···∪ A n = Ω . 3 Partition A 1 , A 2 , ··· , A n is called a partition if the events are mutually exclusive and exhaustive. 4 Complement The complement of event A is the collection of outcomes not in A, i.e. A c = Ω \ A Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2
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Review: Set Theory and Mathematical Theory of Probability Kolmogorov’s Axiom 1 P ( A ) 0 for any event A 2 P (Ω) = 1 3 For any sequence of mutually exclusive events A 1 , A 2 ,... , P [ i = 1 A i ! = i = 1 P ( A i ) (Countable additivity) Mnemonic trick: length of the union of disjoint segments is equal to the sum of their individual lengths. Inclusion-Exclusion Principle P ( A 1 ∪···∪ A n ) = n i = 1 P ( A i ) - i 1 < i 2 P ( A i 1 A i 2 )+ ··· +( - 1 ) n - 1 i 1 < i 2 < ··· < i n P ( A i 1 A i 2 ∩···∩ A i n ) = n j = 1 ( - 1 ) j - 1 i 1 < i 2 < ··· < i j P ( A i 1 A i 2 ∩···∩ A i j ) where n can also be . Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2
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Review Definition of conditional probability For any two events A and B , the conditional probability of A given the occurrence of B is written as P ( A | B ) and is defined as P ( A | B ) = P ( A B ) P ( B ) provided that P ( B ) > 0. Multiplication theorem 1 For any two events A and B with P ( B ) > 0, P ( A B ) = P ( B ) P ( A | B ) . 2 For any three events A , B , C with P ( B C ) > 0, P ( A B C ) = P ( C ) P ( B | C ) P ( A | B C ) . Chan Chi Ho, Chan Tsz Hin & Shi Yun Example class 2
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Review Independence 1 Two events A and B are called independent if and only if P ( A B ) = P ( A ) P ( B ) . If P ( A ) > 0,then A and B are independent if P ( B | A ) = P ( B ) . 2 The events A 1 , A 2 , ··· , A k are (mutually) independent if and only if the probability of the intersection of any combination of them is equal to the product of the probabilities of the corresponding single events.
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