Math 419 – Spring 2016 Common Continuous Random Variables Problem Set 6 1. The losses on a policy are modeled by a uniform random variable on [0, 15000]. If there is a deductible of 1000, find the expected benefit payment. 2. Losses, X, follow a uniform distribution on [0, 2000]. A basic policy reimburses a policy holder for up to 400 of their loss, and if the loss is greater than 400, a supplemental policy reimburses for up to 500 over the reimbursement of the basic policy. What is the expected payout of a supplemental policy to the policyholder? 3. Let X be an exponential random variable with P(X ³ 1) = 0.8. What is F(x) ? 4. Losses are modeled by an exponential distribution with mean 500. Calculate the interquartile range of the losses. 5. Let X be an exponential random variable with mean 9. Find V[X | X > 5]. 6. An actuary models the occurrence of claims from a portfolio of insurance policies. The time until the occurrence of the second claim is modeled by a gamma distribution with mean 10 and variance 50. Given that this time exceeds 10, determine the probability that it exceeds 20. 7. Three spotlights illuminate a billboard. The lifetime of each spotlight is exponential with mean 2 years. Assume the lifetimes of the spotlights are independent. What is the probability that they will all continue to illuminate at the end of 1 year? 8. Consider an insurance policy that reimburses annual hospital charges for an insured individual. The probability of any individual being hospitalized in a year is 0.15. Once an individual is hospitalized, the charges X have a probability density function f(x) = 0.1e – 0.1x for x > 0. We have an insurance pool that reimburses hospital charges for 200 individuals. Assume that claims for each individual are independent of the other individuals. If the insurer includes a deductible of 5 on annual claim payments for each individual, what would the expected claim payments be for the pool? 9. Theft losses are normal with mean 100 and standard deviation 40. Fire losses are normal with mean 150 and standard deviation 30. Assume that theft and fire losses are independent. What is the coefficient of variation of the total losses? 10. The size of a population of a country is currently 8.7 million people. In one year's time the population will be 8.7 x 106 x eX , where X is normally distributed, with mean 0.01 and variance 0.0004. Calculate the probability that the population will increase during the year, but by no more than 300,000. 11. A random variable X is normally distributed with mean 5 and standard deviation 2. What is the probability that 8X – X 2 < 1 ?