AMS410_HWSOL2

# AMS410_HWSOL2 - AMS 410 Actuarial Mathematics Fall 2009...

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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Homework 2 solution: Integrals + Discrete distribution 10/8/2009 1. Evaluate the following integrals: Remember the formulas I gave you in class: • ´ 1 x c dx = 1 c +1 ( c >- 1 ) • ´ ∞ 1 x- c dx = 1 c- 1 ( c > 1 ) • ´ ∞ e- cx dx = 1 c ( c > ) • ´ ∞ a e- cx dx = e- ca c ( c > , a > ) (a) ´ 1 ( x 3 + 2 x 5 + 3 x 10 ) dx ´ 1 ( x 3 + 2 x 5 + 3 x 10 ) dx = 1 1+3 + 2 · 1 1+5 + 3 · 1 1+10 = 0 . 856 (b) ´ ∞ (1 + x )- 5 dx Let y = 1 + x , dx = dy , ´ ∞ (1 + x )- 5 dx = ´ ∞ 1 y- 5 dy = 1 4 (Don't forget to change the upper and lower limits) (c) ´ ∞ x (1 + x )- 5 dx Let y = 1 + x , dx = dy , ´ ∞ x (1 + x )- 5 dx = ´ ∞ 1 ( y- 1) · y- 5 dy = ´ ∞ 1 ( y- 4- y- 5 ) dy = 1 3- 1 4 = 1 12 (d) ´ ∞ 1 e- 3 x dx ´ ∞ 1 e- 3 x dx = e- 3 × 1 3 = 1 3 e- 3 (e) ´ ∞ 1 xe- 3 x dx ´ ∞ 1 xe- 3 x dx = ´ ∞ 1 x- e- 3 x 3 dx = x- e- 3 x 3 | ∞ 1- ´ ∞ 1- e- 3 x 3 dx = lim x →∞ x- e- 3 x 3- 1 · - e- 3 3 + 1 3 ´ ∞ 1 e- 3 x dx = 1 + 1 3 e- 3 + 1 9 e- 3 = 4 9 e- 3 . Notice that lim x →∞ x- e- 3 x 3 = (- 1 3 ) lim x →∞ x e 3 x . Using L'hopital's rule, lim x →∞ x e 3 x = lim...
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AMS410_HWSOL2 - AMS 410 Actuarial Mathematics Fall 2009...

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