FM5002-HW12-5.2.12

# FM5002-HW12-5.2.12 - Financial Mathem atics 5002 Hom ework...

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Financial Mathematics 5002 : Homework 12 (0059, 0060, 0061, 0062) Due on 2 May 2012 Scot Adams Solutions 0059 M 1 .Let C 1 , C 2 , ...be the usual sequence of L P 1 R M valued PCRVs that model coin M flipping. For all integers n G 1, let D n E C 1 p ... p C n . a.Compute lim n RI E l D n S n 4 p D n S n 5 r . b.Let f ± x ² E ³ 4 x, if 1 L x L 7 0, otherwise. Compute lim n E l f D n S n r . c. Compute lim n E l D n S n 4 M D n S n 6 r . d. Compute lim n E l D n S n 9 p D n S n 2 r . a. lim n E l D n S n 4 p D n S n 5 r E 1 2 Π I MI I ± x 4 p x 5 ² e M x^2 s 2 D x E 3. b. lim n E l f D n S n r E 1 2 Π I 1 7 4 xe M s 2 D x E 4 2 Π ´ e M 1 s 2 M e M 49 s 2 µ . c. lim n E l D n S n 4 M D n S n 6 r E 1 2 Π I I ± x 4 M x 6 ² e M s 2 D x E 3 M 15 EM 12. d. lim n E l D n S n 9 p D n S n 2 r E 1 2 Π I I ± x 9 p x 2 ² e M s 2 D x E 1. 0059 − 2 . a.Compute 1 2 Π I I e 2x M 9 e M x 2 s 2 D x. b.Compute 1 2 Π I M 1 2 e 2x M 9 e M x 2 s 2 D

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c.Compute 1 2 Π I MI I L e 2x M 9 M 2 R e M x 2 S 2 D x. d.Compute 1 2 Π I I L e 2x M 9 M 2 R P e M x 2 S 2 D e.Compute 1 2 Π I I L e 2 ix M 9 M 2 R e M x 2 S 2 D a. 1 2 Π I I e 2x M 9 e M x 2 S 2 D x E e M 7 2 Π I I e M x 2 S 2 D x E e M 7 . b. 1 2 Π I M 1 2 e 2x M 9 e M x 2 S 2 D x E e M 7 2 Π I M 1 2 e M x 2 S 2 D x E e M 7 2 Π I M 3 0 e M x 2 S 2 D x E e M 7 L C L 0 R MC L M 3 RR . c. 1 2 Π I I L e 2x M 9 M 2 R e M x 2 S 2 D x E 1 2 Π I I e 2x M 9 e M x 2 S 2 D x M 2 2 Π I I e M x 2 S 2 D x E e M 7 M 2. d. 1 2 Π I I L e 2x M 9 M 2 R P e M x 2 S 2 D x E 1 2 Π I L log L 2 R P 9 RS 2 I L e 2x M 9 M 2 R e M x 2 S 2 D x E 1 2 Π I L log L 2 R P 9 RS 2 I e 2x M 9 e M x 2 S 2 D x M 2 2 Π I M L log L 2 R P 9 RS 2 e M x 2 S 2 D x E e M 7 C LL M log L 2 R P 9 RS 2 P 2 R M 2 C L M L log L 2 R P 9 2 R . e. 1 2 Π I I L e M 9 M 2 R e M x 2 S 2 D x E M 2 P 1 2 Π I I e M 9 e M x 2 S 2 D x E e M 11 M 2. 0059 − 3 . Let X be a binary PCRV such that Pr[X = a] = p and Pr[X = b] = q, where p + q = 1. Let f (t) be the Fourier transform of the distribution of X. a.Compute E l X 4 r . b.Compute E l X 5 r . c. Compute f (0). d. Compute f L 4 R (0). e. Compute f L 5 R (0). a.E l X 4 r E p a 4 P q b 4 . b.E l X 5 r E p a 5 P q b 5 . c.The generating function for the distribution of Xis p z a P q z b , so that the Fourier transform is f L t R E p e M ia t P q e M i b t and f L 0 R E p P q E 1. d.f ’ L t R EM i a e M p M i b e M i b t q, f ’’ L t R E M a 2 e M p M b 2 e M i b t q, 2 FM5002-HW12-5.2.12.nb
f L 3 R L t R E i a 3 e M ia t p P i b 3 e M i b t q, so that f ’’ L 0 R E a 4 p P b 4 q. e.f L 4 R L t R E a 4 e M p P b 4 e M i b t q, so that f L 5 R L t R EM i a 5 p M i b 5 0059 − 4 . Let X be a binary PCRV such that Pr[X = a] = p, Pr[X = b] = q, and Pr[X = c] = r, where p + q + r = 1.

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