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Unformatted text preview: HW1 ST520  Hannig Adam Labadorf ST520  Hannig September 3, 2007 HW1 1. 1.2.2 Definition of a σfield: (a) Ø ∈ F (b) if A 1 ,A 2 ,... ∈ F then S ∞ i =1 A i ∈ F (c) if A ∈ F then A c ∈ F Proofs: (i) A ∩ B : A,B ∈ F by the problem definition, and by definition (c) of σfields we know that A c ,B c ∈ F and by definition (b) we know that A ∪ B ∈ F . By these axioms we state that A c ∪ B c ∈ F . By applying Morgan’s Law, we can show that ( A c ∪ B c ) c = ( A ∩ B ) . Therefore, ( A ∩ B ) ∈ F . (ii) A \ B : By definition A \ B = A ∩ B c . We know that if B ∈ F then B c ∈ F by (c) and that A ∩ B ∈ F by (i). Therefore, A \ B ∈ F . (iii) A Δ B : By definition A Δ B = A ∪ B \ ( A ∩ B ) = A ∪ B ∩ ( A ∩ B ) c . By De Morgan’s Law, A ∪ B ∩ ( A ∩ B ) c = A ∪ B ∩ A c ∪ B c . A ∪ B ∈ F from (b), A c ,B c ∈ F from (c), and A ∩ B ∈ F from (i). Therefore, A Δ B ∈ F . 1.2.4 We show that G is a σfield by proving each condition listed in 1.2.2 separately: (a) Ø ∈ G Let some A ∈ F be such that A / ∈ B . From the definition of G , the set A ∩ B ∈ G and, since they are disjoint, A ∩ B = Ø . Therefore, Ø ∈ G X . (b) if A 1 ,A 2 ,... ∈ G then S ∞ i =1 A i ∈ G Let some series C 1 ,C 2 ,C 3 ,... be such that C 1 = A 1 ∩ B,C 2 = A 2 ∩ B,... . Therefore, C i ∈ G . Since we know that any A 1 ∪ A 2 = A 3 ∈ F , there must be some ( A 1 ∩ B ) ∪ ( A 2 ∩ B ) = ( A 3 ∩ B ) ∈ G by the definition of G . It follows that there is some A 4 such that A 3 ∪ A 4 = A 5 ∈ F and, therefore, ( A 3 ∩ B ) ∪ ( A 4 ∩ B ) = ( A 5 ∩ B ) ∈ G . Thus, S ∞ i =1 A i ∈ G X . (c) if A ∈ G then A c ∈ G Let some C ∈ F . Since F is a σfield we know that C c ∈ F . Let A ∈ G such that C ∩ B = A and C c ∩ B = A c with respect to B . Therefore, A,A c ∈ G X . 1.3.1 To prove that 1 12 ≤ P ( A ∩ B ) ≤ 1 3 we must first show that A and B are not disjoint. We do this by contradicting the axiom: P ∞ [ i =1 A i !...
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This note was uploaded on 06/16/2009 for the course STAT 520 taught by Professor Hannig,j. during the Spring '09 term at Colorado State.
 Spring '09
 Hannig,J.
 Statistics, Probability

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