S490C F07 Final Solutions

# S490C F07 Final Solutions - STAT 490C FINAL December 12,...

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Unformatted text preview: STAT 490C FINAL December 12, 2007 1. A random number of 0.6 is generated from a uniform distribution on (0, 1). Using the inverse transformation method, calculate the simulated value of X assuming: i. (3 points) X is distributed Pareto with ct = 3 and 0 = 2000 0.5x 0§X<l.2 ii. (2 points) F(X) = 0.6 1.2 _<_ X < 2.4 0.5X- 0.6 2.4gx<3.2 iii. (2 points) F(X) = 0.1x -1 10 f X <15 0.05x 15 5 X < 20 SEE MW at W” 2. (7 points) You are given the following random sample of 3 data points from a population with a Pareto distribution with G = 70: X: 15 27 43 Calculate the maximum likelihood estimate for CL. 555. #W Hg (5 points) During a one-year period, the number of accidents per day in the parking lot of the Steenman Steel Factory is distributed: Number of Accidents Days 0 220 l 100 2 30 3 10 4 2 5 0 6 1 7 1 8 1 9+ 0 The accidents are assumed to be distributed following a Poison distribution with )b estimated by the maximum likelihood estimator. Calculate the 95% conﬁdence interval for X. ,‘ . ("a \$5755 “t” “A w W? W (1110313 +(Joe)(> «if r 9K” W X W Mia) May it) if” 5‘”) bi +67%) waxagwm \ M waved/mama»: we“; ﬂ " ”' ﬂﬂy. T grit} (:39 {a g as During a one—year period, the number of accidents per day in the parking lot of the Steenman Steel Factory is distributed: Number of Accidents 2 l 30 I 3 l 10 4+ 5 You are given the following hypothesis: H0: The distribution of the number of accidents per day is distributed as Poison with a mean of 0.625. H1: The distribution of the number of accidents per day is not distributed as Poison With a mean of 0.625. (6 points) Calculate the chi—square statistic. (2 points) Calculate the critical value at a 10% signiﬁcance level. (1 point) State Whether you would reject the H0 at a 10% signiﬁcance level. 5,556“ HW #/‘f/ Based on a random sample, you are testing the following hypothesis: Ho: The data is from a population distributed binomial with m = 6 and q = 0.3. H1: The data is from a population distributed binomial. You are also given: L(60) = .1 and L(01) = .3 (2 points) Calculate the test statistic for the Likelihood Ratio Test (2 points) State the critical value at the 10% signiﬁcance level §>EE 44- /‘/3 (4 points) You are given the following 9 claims: X: 10, 60, 80, 120, 150, 170, 190, 230, 250 The sum ofX : 1260 and the sum of X2 = 227,400. The data is modeled using an exponential distribution With parameters estimated using the percentile matching method. Calculate 9 based on the empirical value of 120. 5% WM #5 (7 points) Craig has an automobile insurance policy. The policy has a deductible of 500 for each claim. The number of claims follows a Poison distribution with a mean of 2. Automobile claims are distributed exponentially with a mean of 1000. The insurance company uses simulation to estimate the claims. A random number is ﬁrst used to calculate the number of claims. Each claim is then estimated using random numbers and the inverse transformation method. The random numbers generated from a uniform distribution on (0, l) are 0.7, 0.1, 0.5, 0.8, 0.3, 0.7, 0.2. Calculate the simulated amount that Craig would have to pay in the first year. (4 points) A sample of two selected from a uniform distribution over (0,U) produces the following values: 3 7 You estimate U as the MaX(X1, X2). Estimate the Mean Square Error of your estimate of U using the bootstrap method. £56 #142 if /5/ 9. Schmidt’s Bakery has workers’ compensation claims during a month of: 100, 350, 550, 1000 Schmidt’s owner, a retired actuary, believes that the claims are distributed exponentially with 0 = 500. He decides to test his hypothesis at a 10% signiﬁcance level. (5 points) Calculate the Kolmogorov-Smirnov test statistic. éﬂg 717‘: JEE (2 point) State the critical value for his test. (1 point) State his conclusion. (1 points) He also tests his hypothesis using the Anderson-Darling test statistic. State the values of this test statistic under Which Mr. Schmidt would reject his hypothesis. 10. (6 points) The following sample is from a population with an exponential distribution: X: 300, 400, 500 The parameter 6 is estimated using the maximum likelihood estimator. Use the delta method to estimate the mean and variance of p = Pr(x>300). 11. (4 points) You are given the information matrix for the estimation of (X and 9 is: 2.00 -0.40 -0.40 1.08 Calculate the Var(0c), the Var(9), and COV(0L,9) 7— Q/QzéD 0 //,//_fs yMA 12. (6 points) You are given the following sample of claims obtained from an inverse gamma distribution: X: 12, 13, 16, 16, 22, 24, 26, 26, 28, 30 The sum ofX is 213 and the sum osz is 4921. Calculate 0L and 9 using the method of moments. Eé 50/ Mg) 13. (3 points) State whether the following are true or false i. The principle of parsimony states that a more complex model is better because it will always match the data better. ii. In judgment-based approaches to determining a model, a modeler’s experience is critical. iii. In most cases, judgment is required in using a score—based approach to selecting a model. £55 AW #M/ ...
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## This note was uploaded on 03/15/2012 for the course STAT 490 taught by Professor Na during the Fall '11 term at Purdue University-West Lafayette.

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S490C F07 Final Solutions - STAT 490C FINAL December 12,...

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