Example_Class_2_(Solution)

Example_Class_2_(Solution) - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 2 Review Set Theory and Mathematical Theory of Probability 1. 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 . 2. Language of Probability (a) Mutually exclusive A 1 ,A 2 , ··· n are mutually exclusive if A i A j = φ for all i 6 = j . (b) Exhaustive A 1 2 , n are exhaustive if A 1 A 2 ∪ ··· ∪ A n = Ω . (c) Partition A 1 2 , n is called a partition if the events are mutually exclusive and exhaustive. (d) Complement The complement of event A is the collection of outcomes not in A, i.e. A c = Ω \ A 3. Kolmogorov’s Axiom (a) P ( A ) 0 for any event A (b) P (Ω) = 1 (c) For any sequence of mutually exclusive events A 1 2 ,... , P [ i =1 A i ! = X 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. 4. Inclusion-Exclusion Principle P ( A 1 A 2 A n ) = n X i =1 P ( A i ) - X i 1 <i 2 P ( A i 1 A i 2 ) + +( - 1) n - 1 X i 1 <i 2 < ··· <i n P ( A i 1 A i 2 ∩ ··· ∩ A i n ) = n X j =1 ( - 1) j - 1 X i 1 <i 2 < ··· <i j P ( A i 1 A i 2 A i j ) where n can also be . 1
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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 ) . 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 , · · · 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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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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Example_Class_2_(Solution) - THE UNIVERSITY OF HONG KONG...

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