FM5002-HW10-4.18.12 (1)

FM5002-HW10-4.18.12 (1) - Financial Mathem atics 5002 Hom...

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Financial Mathematics 5002 : Homework 10 (0050, 0051) Due on 18 April 2012 Scot Adams Solutions 0050 − 1 . Using Stirling’ s formula, find constants C, k, and b such that 7 n 4 n T C L n k R L b n R , that is, such that lim n RI 7 n 4 n L C L n k R L b n RR M 1 E 1. Recall that Stirling’ s formula states that n eT 2 Π n n e n . Then 7 n 4 n E L 7 n R e L 4 n R e L 3 n R e T 7 2 Π n l 7 n e r 7 n 4 2 Π n l 4 n e r 4 n 3 2 Π n l 3 n e r 3 n E 7 L 7 n R 7 n 12 2 Π n L 4 n R 4 n L 3 n R 3 n E 7 24 Π n M 1 S 2 L 7 n R 7 L 4 n R 4 L 3 n R 3 n E 7 24 Π n M 1 S 2 7 5 4 4 3 3 n . Therefore, C E 7 24 Π , k EM 1 S 2, b E 7 5 4 4 3 3 . 0051 M 1 . For all integers n G 1, let H n be the nth histogram and define p n L x R as follows : If xis outside the span of H n , then p n L x R E 0. If xis under exactly one bar of H n , then p n L x R is the height of that bar. If xis at the borderline of two bars, then p n L x R is the height of the leftmost of those two bars.

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