HW1_F09 - Stat219 Math 136 Stochastic Processes Homework...

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Stat219 / Math 136 - Stochastic Processes Homework Set 1, Fall 2009. Due: Wednesday, September 1 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 \ i - 1 j =1 A j . Then the B i are disjoint and we let C = i =1 A i = 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 ). (e) Inclusion-exclusion rule: IP( n i =1 A i ) = i IP( A i ) - i<j IP( A i A j ) + i<j<k IP( A i A j A k ) - · · · + ( - 1) n +1 IP( A 1 ∩ · · · ∩ A n ) .
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