ASM practice exam 1 #19

frisbbzepp · #1 10-22-2006, 01:17 PM

A city purchasesa snow cleaning contract for a winter season. The contract pays the cost of cleaning the snow, subject to an aggregate deductible of 30 for the season. You are given:
(i) The number of snowstorms is binomial with m=10, q=0.3
(ii) The cost of cleaning for each snowstorm has the following distribution:

Pr Amount
0.4 10
0.3 20
0.3 35

Determine the expected aggregate reimbursement the city receives during a season.

I wanted to calculate $E[(S-30)_+]$ using the recursive formula. I got that
$Pr(S=0)=(0.7)^{10}$ ,
$Pr(S=10)=1.2(0.7)^9$ ,
$Pr(S=20)=45(0.3)^2(0.7)^8(0.4)^2+10(0.3)^2(0.7)^9$ , and
$Pr(S=30)=45(0.3)^2(0.7)^8(0.4)(0.3)(2)+120(0.3)^3( 0.7)^7(0.4)^3$ .

Then I wanted to do $E[(S-10)_+]=E(S)-[1-F_S(10)]$ , and so on. I'm not getting the answer that the book is getting (I think I need to find $E(S\wedge30)$ . Can anyone point out my error or maybe spell out the solution using this method? Thanks.

(As an aside, does anyone know how to type combinations using MimeTex? I tried \binom but that didn't work.)

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