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# HW3 - AMS 410 Actuarial Mathematics Fall 2009 Homework 3...

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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Homework 3: Continuous distribution Due: 10/20/2009 1. The number of claims that an insurance company gets is a random variable, and the probability of getting k claims is 1k8 0.15k 0.85(18−k) , k = 0, 1, · · · , 18. What's the mode of the distribution? 2. The moment generating function of a random variable X is MX (t) = t < λ. What's V ar(X )? λ λ−t r , λ > 0, r > 0, 3. An insurer's annual weather related-loss, X , is a random variable with density function 2.5(200)2.5 for x > 200 x3.5 . Calculate the dierence between the 30th percentile and f ( x) = 0 otherwise 70th percentile. 4. Suppose X has exponential distribution with median 3. Determine: (a) E (X ) (b) The 75th percentile of the distribution of X 5. Suppose X has exponential distribution with mean 3. Determine: (a) The median (b) The 75th percentile of the distribution of X 6. An insurance company's monthly claims are modeled by a continuous, positive random variable X , whose probability density function is proportional to (1 + x)=4 , where 0 < x < 1. Determine the company's expected monthly claims. 7. The lifetime of a light bulb follows exponential distribution with a mean of 100 days. Find the probability that the light bulb's life: (a) Exceeds 100 days, (b) Exceeds 400 days given it exceeds 100 days. 8. An insurance company insures a large number of homes. The insured value, X , of a randomly 3x−4 for x > 1 selected home is assumed to follow a distribution with density function f (x) = . 0 otherwise Given that a randomly selected home is insured for at least 1.5, what is the probability that it is insured for less than 2? 9. The pdf of a random variable Y is f (y ) = 3|2 < Y < 4]. y 6 − y2 36 0 for 0 < y < 6 . Calculate P [1 < Y < otherwise 10. Suppose X has a uniform distribution on [0,100], calculate the expectation of X given that X > 50. 1 ...
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