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Unformatted text preview: Stat 479
Final
Fall 2009 1. You are given the following data which is assumed to be drawn from a uniform distribution over the
range of 0 to co: .
_ Determine maximum likelihood estimate of a). If you knew that the 10 largest claims ranged from 5,150 to 8000, what would be the maximum
likelihood estimate of 03? 2. You are given the following sample taken from distribution that follows a Gamma distribution:
8 12 15 21 Using the Method of Moments, estimate the parameters of the Gamma distribution. 3. A sample of claims from Cinﬁo Casualty Insurance Company is listed below:
500, 600, 800, 1000, 1300, 1800, 2400, 2800, 3000, 5000 Frank, the Company’s actuary, uses the method of percentile matching at the 80th percentile to
determine the parameter 0 for an Inverse Pareto distribution assuming that T = 2. Determine Frank’s estimate of 0. 4. Nick and Vaidehi are covered by a liability insurance policy.
The number of claims for Nick and Vaidehi is assumed to follow a Poison distribution with 9» = 2. The amount of each claim for Nick has the following cdf: 0.001x 0 g x < 550
F(x) =
0.001x+0.15 5505x5850 The amount of each claim for Vaidehi has the following cdf: 0.001x 0 5 X < 550
F(x) = 0.55 550§x<750
0.001x — 0.20 750 5 x 31200 An insurance company uses simulation to estimate the total amount of claims to be incurred by Nick
and Vaidehi. First, the company determines the number of claims for Nick and then the amount of
each claim for Nick. Next, the company determines the number of claims for Vaidehi. Finally, they
simulate the amount of each of Vaidehi ‘5 claims. The random numbers generated for the simulation are: 0.50 0.30 0.65 0.75 0.70 0.55 0.25 0.60 0.45 0.15 Determine the total amount claims paid for each of Nick and Vaidehi. 5. To estimate E(X), you have simulated ﬁve observations from the random variable X.
The values are l, 3, 5, 8, and 8.
Your goal is to have the standard deviation of the estimate of E(X) be less than 0.5. Estimate the total number of simulations needed. 6. The mean of a distribution will be estimated by taking the average of the smallest value and the
largest value from a sample. In other words, [:2 = X2, .. ., X“) + MaX(X1, X2, . . ., The following sample is selected from the distribution:
4, 4, 7 Using the bootstrap method, estimate the mean square error in the above estimator. 7. An insurance policy pays claims up to a limit of 2000. A random sample of two payments is
obtained as follows: 300 and 2000. The claims are assumed to follow an Inverse Pareto distribution with r = 1. Calculate the maximum likelihood estimate for 0. 8. Claims for a hospital stay are assumed to follow an exponential distribution.
Hong Insurance offers a hospitalization policy that covers hospital claims up to a maximum of 10,000. A sample of ten claims is drawn to estimate 0. The payments on the ten claims in the
sample are: 500 700 850 1200 1500 2500 5000 7800 10,00010,000 Estimate 0 using the maximum likelihood estimate. 9. You have a sample of 19 claims for dental insurance claims. Using the smoothed empirical
distribution, the 42nd percentile is 207.4 and the 44th percentile is 215.8. Determine X3 and X9. 10. During a oneyear period, Ren Wreckers suffered the following liability claims:
400 900 1400 3300
The losses are assumed to be distributed following a Pareto distribution with a = 4 and 0 = 5000. Xuelu wants to test this hypothesis using the KolmogorovSmirnov test with a signiﬁcance level of
5%. Calculate the Kolmogorov—Smimov test statistic. State the critical value for the test. State your conclusion. 11. During a one—year period, Ren Wreckers suffered the following liability claims:
400 900 1400 3300
The losses are assumed to be distributed following an Pareto distribution with 0'. = 4 and 0 = 5000.
You want to create app plot to evaluate the assumption. Determine the plotted point for the observed value of 900. 12. During an hour, the number of cars that arrive at a gas station during any one minute is given below: Number of Cars Arrivin Assuming that arrivals are distributed as a Poisson distribution, determine the 90% linear conﬁdence
interval for k, the expected number of cars arriving at the gas station in any one minute period. 13. During the last year, Edson Fire Insurance Company incurred 20 claims. The claims were
distributed as follows: The Company wants to test their belief that the claims are distributed exponentially with a mean of
20,000. Therefore, H0: Fire Claims are distributed exponentially with a mean of 20,000.
H1: Fire Claims are not distrusted exponentially with a mean of 20,000. Calculate the Chi Square test statistic. Calculate the critical values for the Chi Square test at a 5% signiﬁcance level. State your conclusion with regard to Edson’s hypothesis. 14. A golf course buys an insurance policy from Amstutz Assurance Company which covers damage to
adjacent houses caused by errant golf balls. The policy covers each claim after a deductible of 50
for each claim. The number of claims during a round of golf follows a binomial distribution with m = 3 and q = 0.6
The amount of each claim follows a Pareto distribution with on = 11 and 0 = 1000. Using simulation, Alex wants to estimate the total claims that will need to be paid by Amstutz
Assurance Company. He does so by estimating the claims for each golfer. John and Ian are the ﬁrst
two golfers. First, Alex determines the number of claims for John and then the amount of each
claim for John. Next, Alex determines the number of claims for Ian. Finally, Alex simulates the
amount of each of Ian‘s claims. The random numbers used in the simulation are: 0.10 0.85 0.50 0.30 0.90 0.45 0.60 0.95 0.75 0.05 Calculate the amount the Amstutz will have to pay for as a result of Ian’s round of golf. 15. You are given the following sample taken from distribution that follows a Binomial distribution:
6 7 9 ll 12 Using the Method of Moments, estimate the parameters of the Binomial distribution. ...
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This note was uploaded on 03/15/2012 for the course STAT 479 taught by Professor Na during the Spring '10 term at Purdue.
 Spring '10
 NA

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