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FM5002-HW8-3.28.12

# FM5002-HW8-3.28.12 - Financial Mathem atics 5002 Hom ework...

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Financial Mathematics 5002 : Homework 8 (0047-0048) Due on 28 March 2012 Scot Adams Solutions 0047 − 1 . Suppose Pr[A | B] = 0.6, Pr[A] = 0.3, and Pr[B] = 0.5. a. Find Pr[A and B]. b. Find Pr[B | A]. a. Pr L A B R E Pr L Aand B R L B R , so that Pr L R E 0.3. b. Pr L B A R E L R L A R E 1. 2 . a. Find two PCRVs X and Y such that Pr[(X = 1) | (Y = 2)] = 0.6, Pr[X = 1] = 0.3, and Pr[Y = 2] = 0.5. b.Compute Pr[(Y = 2) | (X = 1)]. a. Let X l t r E ± 1 if 0 L t L 0.3 0 if 0.3 l t L 1 and Y l t r E ± 2 if 0 L t L 0.5 0 if 0.5 L t L 1 . Then Pr Ll X E 1 r l Y E 2 rR E P X E 1 and Y E 2 rR P L Y E 2 R E 0.3 0.5 E 0.6. b. Pr Y E 2 r l X E 1 rR E P X E 1 and Y E 2 rR P L X E 1 R E 0.3 0.3 E 1. 0047 M 3 . Let C 1 , C 2 , C 3 , ... be our standard sequence of coin M flipping PCRVs. For all integers n G 1, let D n E C 1 P ... P C n n . a. Compute lim n RI E L D n 6 R . b. Compute lim n E L 80 l e 4 D n M 3 M e r P R . a. Recall that 1 2 Π I MI I x 2 n e M x 2 2 D x E l 2 n M 1 r ee . By the Central Limit Theorem, lim n E L D n 6 R E 1 2 Π I I x 6 e M x 2 2 D x E 15.

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b. Again, by the Central Limit Theorem, lim n RI E L 80 l e 4 D n M 3 M e R P r E 80 2 Π I MI I l e 4 x M 3 M e R P e M x 2 2 D x E 80 2 Π I 1 I l e 4 x M 3 M e R e M x 2 2 D x E 80 e M 3 2 Π I 1 I l e 4 x R e M x 2 2 D x M 80 e 2 Π I 1 I e M x 2 2 D x E 80 l e 5 C l 3 R M e C l M 1 RR . 0047 − 4 . Let X and Y be PCRVs such that Pr[X = 4] = 0.8, Pr[(X = 4) and (Y = 9)] = 0.35, and Pr[(X = 4) and (Y = 2)] = 0.45. a. Find Pr[(Y = 2) | (X = 4)].
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