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A life insurer has created a special one-year term insurance policy for a pair of business people who travel to high risk locations. The insurance policy pays nothing if neither die in the year, $\$100,000$ if exactly one of the two dies, and $\$K > 0$ if both die. The insurer determines that there is a probability $0.1$ that at least one of the two will die during the year and a probability of $0.08$ that exactly one of the two will die during the year. You are told that the standard deviation of the payout is $\$74,000$. Find the expected payout for the year on this policy.

This what I did: Let $X$ be the total payout for the victims. $$\mathbb{E}[X]= P(A\text{ dies})*10,000 + P(B\text{ dies})*10,000 + P(\text{none dies})*0 + P(\text{both die})*K.$$ So if I plug in all the values, will my answer be correct?

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Note that, as far as the insurance company is concerned, there are only three distinct outcomes:

  1. Neither of them dies. This has probability $0.9$ (why?), and will result in a payout of $\$0.$
  2. Exactly one of them dies. This has probability $0.08,$ and will result in a payout of $\$100,000.$
  3. Both of them die. This has probability $0.02$ (why?), and will result in a payout of $\$K.$

The expected payout (in dollars) is then $$\Bbb E[X]=0.9\cdot 0+0.08\cdot100,000+0.02\cdot K=8,000+0.02K.$$

Now, we also see that $$\Bbb E[X^2]=0.9\cdot 0^2+0.08\cdot100,000^2+0.02\cdot K^2=800,000,000+0.02K^2,$$ so we can use the standard deviation $$\sigma=\sqrt{\Bbb E[X^2]-\Bbb E[X]^2}$$ to solve for $K,$ and hence find the expected payout.

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