Test 4 Solutions

Test 4 Solutions - STAT 230 Test 4 March 7, 2007 3:30 pm...

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STAT 230 Test 4 March 7, 2007 3:30 pm 4:10 pm 1. An insurance company sells a one-year automobile policy at a fixed premium rate R . The total amount of loss, X , incurred from a particular policy holder and paid by the insurance company, is a random variable with possible values 0, 1, 2, 3, 4 or 5 (in thousand dollars). It is known that 95 . 0 ) 0 ( = = X P and x k x X P / ) ( = = for = x 1, 2, 3, 4 and 5, where is a constant. k [4] (a) Show that the constant k 0.022 . Solution: We must have 05 . 0 ) 5 1 4 1 3 1 2 1 1 ( = + + + + k , which gives 022 . 0 k . [4] (b) Find the expectation and the variance . ) ( X E ) ( X Var Solution: ; Va 11 . 0 5 ) ( = k X E r ( X ) = 5 k (3 5 k ) 0.318 . [2] (c) The net profit is calculated as X R Y = , where R is the fixed premium rate. Find the expectation and the variance when ) ( Y E ) ( Y Var 20 . 0 = R . Solution: 09 . 0 11 . 0 20 . 0 ) ( ) ( = = X E R Y E ; Var ( Y ) = Var ( X ) 0.318. [2] (d) Suppose the company sells a new policy with a deductible of 2. That is, the insurance company will have zero loss ( 0 = L ) if 2 X ; the company’s loss is 2 = x L if and x X = = x 3, 4 or 5. Find the company’s expected loss under the new policy, assuming the policyholder’s loss, ) ( L E X , follows the same distribution as before. Solution: 0315 . 0 ) 2 ( ) ( 5 3 = = x k x L E n .
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2. Aidan, a construction worker who has been unemployed for last a few years, is just employed by JBC Construction company to work on an apartment building, which is scheduled to start on April 1. If it rains on April 1, the work will be postponed until the first non-raining day after April 1. It is assumed that the probability of raining on any given day is 0.3, independent among different days. Let X be the number of days the work is postponed ( X
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Test 4 Solutions - STAT 230 Test 4 March 7, 2007 3:30 pm...

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