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S479 S12 Homework

# S479 S12 Homework - Homework Problems Stat 479 Chapter 2 1...

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Homework Problems Stat 479 March 6, 2012 Chapter 2 1. Model 1 is a uniform distribution from 0 to 100. Determine the table entries for a generalized uniform distribution covering the range from a to b where a < b. 2. Let X be a discrete random variable with probability function p( x ) = 2(1/3) x for x = 1, 2, 3, … What is the probability that X is odd? 3. * For a distribution where x > 2, you are given: The hazard rate function: h(x) = z 2 /2 x , for x > 2 F(5) = 0.84. Calculate z . 4. F X (t) = (t 2 -1)/9999 for 1<t<100. Calculate f X (50). 5. You are given that the random variable X is distributed as a Weibull distribution with parameters θ = 3 and τ = 0.5. Calculate: a. Pr[X < 5] b. Pr[3 < X < 5] 6. (Spreadsheet Problem) You are given that the random variable X is distributed as a Geometric distribution with paramet ers β = 3. Calculate: a. Pr[X < 5] b. Pr[3 < X < 5] 7. A Weibull Distribution with parameter τ = 1 becomes what distribution? 8. A random variable X has a density function f(x) = 4x(1+x 2 ) -3 , for x > 0. Determine the mode of X. Chapter 3 9. Determine the following for a generalized uniform distribution covering the range from a to b where a < d < b: a. E[X k ] b. E[X] c. Var(X) d. e(d) e. VaR p f. TVaR p 10. For the Pareto distribution, determine E[X], Var(X), and the coefficient of variation in terms of α and θ.

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Homework Problems Stat 479 March 6, 2012 11. For the Gamma distribution, determine E[X], Var(X), the coefficient of variation, and skewness in terms of α and θ. 12. For the Exponential distribution, determine E[X], Var(X), and e(d) in terms of θ. 13. You are given: x 3 /27, for 0 < x < 3 F( x ) = 1, for x >3 Calculate: a. E[X] b. Var(X) c. e(1) d. E[(X-1) + ] e. E[X Λ 2] f. The Median g. The standard deviation principle with k = 1 h. VaR .80 i. TVaR .80 14. (Spreadsheet) If you roll two fair die, X is the sum of the dice. Calculate: a. E[X] b. Var(X) c. e(4) d. E[(X-4) + ] e. E[X Λ 10] f. The Mode g. π 20 15. You are given a sample of 2, 2, 3, 5, 8. For this empirical distribution, determine: a. The mean b. The variance c. The standard deviation d. The coefficient of variation e. The skewness f. The kurtosis g. The probability generating function h. VaR .80 i. TVaR .80
Homework Problems Stat 479 March 6, 2012 16. The random variable X is distributed e xponentially with θ = 0.2. Calculate E[ e 2X ]. 17. * Losses follow a Pareto distribution has parameters of α = 7 and θ = 10,000. Calculate e(5,000) 18. The amount of an individual claim has a Pareto distribution with θ = 8000 and α = 9. Use the central limit theorem to approximate the probability that the sum of 500 independent claims will exceed 550,000. Chapter 4

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