4 An insurance agent sells homeowners insurance policies: basic and supplemental

Question

4 An insurance agent sells homeowners insurance policies: basic and supplemental. Basic homeowners' insurance covers damages to a home and liability up to a pre-specified loss limit. On a given day, this agent sells 1, 2, or 3 basic policies. When selling a policy to a customer, the agent tries to also sell supplementary liability coverage. A customer can only purchase a supplementary liability coverage policy if they purchase a basic homeowners' policy Let Y, represent the number of basic policies sold on a given day and let Y2 represent the number of supplementary policies sold on a given day P(Y| = 1, Y2-1)=0.15 P(Y 2, Y2 1) 0.18 P(Y| = 3, Y,-1) = 0.23 P(Y1-2, Y,-2) = 0.16 P(Y| = 3, Y, = 2) = 0.20 P(Y1-3, Y2-3) = 0.08 (a) Determine E(Y Y2 1) (b) Calculate Var(Y2 Y1 2) (6 pts) (7 pts)

Explanation / Answer

following probability distribution has been given

(a) E(Y1|Y2=1)=sum(Y1*P(Y1|Y2=1) =0.15*1+0.18*2+0.23*3=1.2

(b)V(Y2|Y1=2)=E(Y22|Y1=2)-(E(Y2|Y1=2))2=0.82-0.5=0.32

E(Y22|Y1=2)=1*1*0.18+2*2*0.16+3*3*0=0.82

E(Y2|Y1=2)=1*0.18+2*0.16+3*0=0.5

Y1 1 2 3 total Y2 1 0.15 0.18 0.23 0.56 2 0 0.16 0.2 0.36 3 0 0 0.08 0.08 total 0.15 0.34 0.51 1

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