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Probability : Insurance Policy

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Someone purchases a liability insurance policy, and the probability that they will make a claim on that policy is equal to 0.1. The insurance payout is the amount of money the insurance company must pay if the holder of the policy files a claim. So, if no claim is made, then the insurance payout is just $0; however, whenever a claim is made, the amount of money on the claim, F, has an exponential distribution with mean S 1,000,000. If P is less than the maximum insurance policy payout of $2,000,000, then the insurance payout is equal to F, otherwise, the insurance payout is equal to S2000,000. What is the expected value of the insurance payout? Hint: Consider the indicator of the event "A claim is filed" and how it can be used to compute the expectation in question.

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The solution describes the method of determination of the expected value of the insurance payout using mixture probability distribution.

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Let X denotes the random variable representing the insurance payout.
Then X = 0 if no claim is made
= min{P, } if a claim is made
where P is the amount of ...

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