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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,    ifEdoes not occurA,    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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Most popular questions from this chapter

Aand Bwill take the same 10-question examination. Each question will be answered correctly by Awith probability.7, independently of her results on other questions. Each question will be answered correctly by B with probability .4, independently both of her results on the other questions and on the performance of A.

(a) Find the expected number of questions that are answered correctly by both A and B.

(b) Find the variance of the number of questions that are answered correctly by either A or B

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability .3,and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs \(1000, or the standard model, which costs \)500. Determine the probability mass function of X, the total dollar value of all sales

If E[X] = 1 and Var(X) = 5, find

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(b) Var(4 + 3X).

An urn has n white and m black balls. Balls are randomly withdrawn, without replacement, until a total of k,k…nwhite balls have been withdrawn. The random variable Xequal to the total number of balls that are withdrawn is said to be a negative hypergeometric random variable.

(a) Explain how such a random variable differs from a negative binomial random variable.

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