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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009
Exercise 8: Transformation 11/17/2009
1. Transformation. In the following three cases, calculate the probability density function of Y . •X •X •X follows Uniform (-1,1), follows Uniform (-1,1), follows Uniform (0,1), Y = X 2. Y = X 3. Y = X 2. 2. Transformation. An actuary models the lifetime of a device using the random variable Y = 10X 0.8 , where X is an exponential random variable with mean 1 year. Determine the probability density function f (y ), for y > 0, of the random variable Y . 1 3. Max. Claim amounts for wind damage to insured homes are independent random variables with common density function f (x) =
where 3x−4 0 for x > 1, otherwise. x is the amount of a claim in thousands. Suppose 3 such claims will be made. What is the expected value of the largest of the three claims? 4. Convolution. Two independent random variables X1 and butions with means of 2 and 3 respectively. What the pdf of X2 follow the exponential Y = X1 + X2 ? distri- 5. Variance. Z are independent random variables following exponential distribuα, β and 4. Given the information: U = X + Y + Z , V = X − Y , E (U ) = V ar(V ), E (U ) − E (V ) = V ar(U )−V ar(V ) . What is α? 2
tions with means of X, Y and 2 ...
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- Fall '08