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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 oneyear 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 \
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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 KolmogorovSmirnov 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 AndersonDarling 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 judgmentbased 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 UniversityWest Lafayette.
 Fall '11
 NA

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