Question

1. A large insurance agency services a number of customers who have purchased both a homeowner's policy and an automobile policy from the agency. For each type of policy, a deductible amount must be specified. For an automobile policy, the choices are $100 and $250, whereas for a homeowner's policy, the choices are $0, $100, and $200. Suppose an individual with both types of policies is selected at random from the agency's files. Let X denote the deductible amount on the auto policy and Y denote the deductible amount on the homeowner's policy. The following joint pmf specifies the probabilities associated with each of these pairs: P(x, y) у 100 0 200 X 100 250 .20 .05 .10 .15 .20 .30 a) Find the probability of a $100 deductible on both policies. [1] b) Find the probability of a deductible amount > 100 for a homeowner's policy [2] c) Find the probability of a $100 deductible on the auto policy given a $100 deductible on the homeowner's policy d) Find COV(X,Y) [2]

Answer 1

X: Deductable amount Y: Deductible the auto policy (in dollar) homeowner's policylin dollar). u p(x, y) = P(x=x, Y=y 200 IP[A-a] 100 10-20 0-10 0 - 2010.50 0-3010.50 y 2500-05 0-15 0-25 0.50 1 both policies P[x=x] = { prx,y) P[1-4]0-25 (a) Per $100 deductible on =P [x=400, Y =400) = $(doo 400) = 0-40 (sus) (b) Ruſ deduetable amounts #10 for home owers folicy | - [140] [4.400 +P6=240 - -5 +0. P - 05 ( đ) (C) Pe [ + foo deduetible on auto policy gien e too deductible home owner's policy P[x = 400-too] - P[w-teo, Y=100 on PCX-too] 40 (tre) - 0.10 0.25 96

x . Elxy)=5 Exypla, y = 0 of (100x100 x 0-10) (d) E(X)=5 X-P[x= x] (100 x10-50) +(25040-50) = 175 ECY) Ç y P[Y=y] 0+(10040-25) +(200 40-50) 125 a y + Geox200 40-20) +(950x100x6-15) +(2 50420020-30) 4 = 23,750. -- Cov(x, y) = E(-y) - E(X) E0) = 23450 - (445x425) - 1875 (ste)