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HomeWork_01 - 2. For events E 1 , E 2 , ··· , E n , show...

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Notation: You can either use A 0 (as in class) or A c (as in textbook) to denote the complement of set A . For two events E and F , E F (as in class) or EF (as in textbook) can both be used to denote the intersection. 0. Review the note first. Then read the textbook throughly, and focus on the examples I didn’t talk about in class. 1. Prove the following formula using induction: P ( E 1 E 2 ∪···∪ E n ) = X i P ( E i ) - X i<j P ( E i E j ) + X i<j<k P ( E i E j E k ) - X i<j<k<l P ( E i E j E k E l ) + ··· + ( - 1) n +1 P ( E 1 E 2 ∩ ··· ∩ E n ) . hint: The case n = 3 has been proved in the textbook.
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Unformatted text preview: 2. For events E 1 , E 2 , ··· , E n , show that p ( E 1 ∩ E 2 ∩···∩ E n ) = P ( E 1 ) p ( E 2 | E 1 ) P ( E 3 | E 1 ∩ E 2 ) ··· P ( E n | E 1 ∩···∩ E n-1 ) . 3. Prove Boole’s inequality: P ( ∪ n i =1 E i ) ≤ n X i =1 P ( E i ) . hint: First consider the simple case when n = 2. The idea is to try to partition ∪ n i =1 E i into disjoint set F i ’s, where F 1 = E 1 and F i = E i ∩ ( ∩ i-1 j =1 E j ). 1...
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This note was uploaded on 01/02/2012 for the course MATH 2603 taught by Professor Han during the Spring '10 term at HKU.

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