Credit scores rate the quality of a borrower, suggesting the chances that a borrower will repay a loan. Borrowers with higher scores are more likely to repay loans on time. Borrowers with lower scores are more likely to default. For example, those with scores below 600 have missed 2.5 payments on average in the last six months, compared to almost no late payments for those with scores above 700. A credit score below 650 is often considered subprime. Borrowers with lower ratings have to pay a higher rate of interest to obtain a loan.
Credit scores were once only used to determine the interest rate on loans and decide whether a borrower was qualified for a $200,000 mortgage, for example. Recently, other businesses have begun to use them as well. Some auto insurance policies are more expensive for drivers with lower credit scores because the companies have found that drivers with lower scores are more risky when it comes to accidents as well.
An insurance agent has just opened an office in a community. Conversations with the local bank produced the distribution of scores shown in this table. The table also shows the cost of car insurance.

Motivation
(a) The insurance agent is paid by commission, earning 10% of the annual premium. Why should the agent care about the expected value of a random variable?
(b) Why should the agent also care about the variance and standard deviation?
Method
(c) Identify the random variable described in this table that is most relevant to the agent.
(d) How is this random variable related to his commission?
Mechanics
(e) Graph the probability distribution of the random variable of interest to the agent.
(f) Find the expected commission earned by the agent for writing one policy.
(g) Find the variance and standard deviation of the commission earned by the agent for writing one policy.
Message
(h) Summarize your results for the agent in everyday language.
(i) Do you have any advice for the insurance company based on these calculations?

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Annual Credit Score of Interest Premium for Probability Borrower Rate Car Insurance of Score 700-850 650-699 600-649 500-599 10% Below 500 14% Annual 6% 7% 8% $500 $600 $800 $1,100 $1,500 0.60 0.10 0.15 0.10 0.05