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# sol1 - Stat219 Math 136 Stochastic Processes Homework Set 1...

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Unformatted text preview: Stat219 / Math 136 - Stochastic Processes Homework Set 1, Fall 2008. Due: Wednesday, October 1 For questions on grading, see TBA 1. Exercise 1.1.3 . Let (Ω , F , IP) be a probability space and A,B,A i events in F . Prove the following properties of IP. (a) Monotonicity. If A ⊆ B then IP( A ) ≤ IP( B ). ANS: A ⊆ B implies that B = A ∪ ( B \ A ). Hence, IP( B ) = IP( A ) + IP( B \ A ). Thus since IP( B \ A ) ≥ 0, we get IP( A ) ≤ IP( B ). (b) Subadditivity. If A ⊆ ∪ i A i then IP( A ) ≤ ∑ i IP( A i ). ANS: For each i set B i = A i \ uniontext i- 1 j =1 A j . Then the B i are disjoint and we let C = uniontext ∞ i =1 A i = uniontext ∞ i =1 B i . Since A ⊆ C , from part (a), IP( A ) ≤ IP( C ) . Also, IP( C ) = ∑ ∞ i =1 IP( B i ) and B i ⊆ A i therefore IP( B i ) ≤ IP( A i ) so IP( C ) ≤ ∑ ∞ i =1 IP( A i ) and hence IP( A ) ≤ ∑ ∞ i =1 IP( A i ). (c) Continuity from below: If A i ↑ A , that is, A 1 ⊆ A 2 ⊆ ... and ∪ i A i = A , then IP( A i ) ↑ IP( A ). ANS: Construct the disjoint sets B 1 = A 1 and B i = A i \ A i- 1 for i ≥ 2, noting that A i = ∪ j ≤ i B j and A = ∪ j B j . Therefore, IP( A i ) = ∑ i j =1 IP( B j ) ↑ ∑ ∞ j =1 IP( B j ) = IP( ∪ j B j ) = IP( A ). (d) Continuity from above: If A i ↓ A , that is, A 1 ⊇ A 2 ⊇ ... and ∩ i A i = A , then IP( A i ) ↓ IP( A ). ANS: Apply part (c) to the sets A c i ↑ A c to have that 1 − IP( A i ) = IP( A c i ) ↑ IP( A c ) = 1 − IP( A )....
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sol1 - Stat219 Math 136 Stochastic Processes Homework Set 1...

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