Hw1sol - HW1 ST520 Hannig Adam Labadorf ST520 Hannig September 3 2007 HW1 1 1.2.2 Definition of a σ-field(a Ø ∈ F(b if A 1,A 2 ∈ F then S ∞

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: HW1 ST520 - Hannig Adam Labadorf ST520 - Hannig September 3, 2007 HW1 1. 1.2.2 Definition of a σ-field: (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 σ-fields 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 σ-field 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 σ-field 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 !...
View Full Document

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.

Page1 / 5

Hw1sol - HW1 ST520 Hannig Adam Labadorf ST520 Hannig September 3 2007 HW1 1 1.2.2 Definition of a σ-field(a Ø ∈ F(b if A 1,A 2 ∈ F then S ∞

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online