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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Exercise 9: Normal Distribution + CLT 12/3/2009
1. Normal. A random variable follows a normal distribution with a mean of 6.72. The 80 percentile of the random variable is 8.4. What's its 90 percentile? 2. Normal. Claims led under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000. What is the probability that the average of 25 randomly selected claims exceeds 20,000 ? 3. Normal. There are two independent random variables: W and X, both normally dis- tributed with the same mean. The variances of W and X are 4 and 12 respectively. What is P [|W − X | < 1]? (in HW5) 1 4. CLT, sum. The total claim amount for a health insurance policy follows a distribution with density function f ( x) = 1 −x/1000 , 1000 e for x > 0. The premium for the policy is set at 100 over the expected total claim amount. If 100 policies are sold, what is the approximate probability that the insurance company will have claims exceeding the premiums collected? 5. CLT, sample size unknown. A company manufactures a brand of light bulb with a life- time in months that is normally distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have independent lifetimes. What is the smallest number of bulbs to be purchased so that the succession of light bulbs produces light for at least 40 months with probability at least 0.9772 ? 2 ...
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