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An insurance company provides policies for a rare risk. For a single customer, the payout \(X\) is a random variable with:
Mean \(\mu = \dollar 500\) (Expected loss per customer).
Standard Deviation \(\sigma = \dollar 10,000\) (Very high volatility because most payouts are 0, but some are huge).
The company has \(n\) customers. Let \(\overline{X}_n\) be the average payout per customer.
If the company has only \(n=1\) customer, what is the standard deviation of the average payout? Is this risky?
If the company has \(n=10,000\) customers, calculate the standard deviation of the average payout, \(\sigma(\overline{X}_{10000})\).
Using the Law of Large Numbers, explain why the insurance business model works despite individual outcomes being unpredictable.
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