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Chapter 4: Q. 4.27 (page 165)

An insurance company writes a policy to the effect that an amount of money Amust be paid if some event Eoccurs within a year. If the company estimates that Ewill occur within a year with probability p, what should it charge the customer in order that its expected profit will be 10 percent ofA ?

Short Answer

Expert verified

The insurance company should charge the customer an amount of A(p+0.1).

Step by step solution

01

Given information

An insurance company writes a policy to the effect that an amount of money A must be paid if some event E occurs within a year.

02

Explanation

Let Xbe a random variable that represents the amount to be paid to the customer.

X=0,鈥呪赌呪赌呪赌�ifEdoes not occurA,鈥呪赌呪赌呪赌�ifEoccurs

Since P(X=0)=PEc=1-pand P(X=A)=P(E)=p, the distribution of Xis

X~0A1-pp

and its expectation is

E[X]=(1-p)0+pA=pA

03

Final answer

Let's say that the company charges the customer an amount of money B. The profit is then B-X, so expected profit is E[B-X]=0.1A. By using additivity of expectation we have

E[B-X]=B-E[X]=B-pA=0.1A

which gives

B=A(p+0.1)

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