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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009
Exercise 6: Other questions 10/20/2009
1. The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least ve hours. 2. An insurance policy is written to cover a loss, X , where X has a uniform distribution on [0, 1000]. At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible? 3. The warranty on a machine species that it will be replaced at failure or age 4, whichever 1 for 0 < x < 5 occurs rst. The machines age at failure, X , has density function f (x) = 5 . 0 otherwise Let Y be the age of the machine at the time of replacement. Determine the variance of Y . 4. Three individuals are running a one kilometer race. The completion time for each individual is a random variable. Xi is the completion time, in minutes, for person i. X1 : uniform distribution on the interval [2.9, 3.1] X2 : uniform distribution on the interval [2.7, 3.1] X3 : uniform distribution on the interval [2.9, 3.3] . The three completion times are independent of one another. Find the probability that the earliest completion time is less than 3 minutes. 1 5. The number of a major hurricane in each year follows a Poisson distribution with parameter λ. It is found that it is 1.5 times as likely that a major hurricane will occur in the next ten years as it is that the next major hurricane will occur in the next ve years. Find λ. 6. Suppose X has the pdf f (x) = 2x, 0 < x < 1. Calculate: P (X > 0.5|X > 0.4), P (X > 0.4|X > 0.5), E (X |X > 0.5). 7. Let X be exponentially distributed with mean 2. Determine: P (X > 5|X > 2), E (X |X > 2), V ar(X |X > 2). 2 ...
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- Fall '08
- Normal Distribution, Probability theory, Exponential distribution, electronic device, completion time