8/13/2019 Old Math 478 Exam 2

1/14

1

To Be Used as a Study Aid for Exam # 2, Math 478 / 568, Spring 2013

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

Actuarial Science Program

DEPARTMENT OF MATHEMATICS

Math 478 / 568 Prof. Rick GorvettActuarial Modeling Spring, 2009

Exam # 2 (17 Problems Max possible points = 40)Thursday, March 19, 2009

You have 75 minutes (from 8:30 to 9:45 am) to complete this exam. The exam is closed-book,closed-note, except that you may refer to your two-sided 3x5-inch notecard. A standard normaldistribution table and other distribution information is provided at the end of this exam (feel free totear it off the back of the exam for convenience).

Problems (1) through (14) are multiple choice, each worth two points. Circle the letter associatedwith thebestanswer to each question.

Problems (15) through (17) are each worth four points. Please provide dollar answers to the nearestcent ($x,xxx.xx), and proportion and probability answers either as percentages to two decimal places(xx.xx%) or as numbers to four decimal places (0.xxxx).

When using the normal distribution, there is no need to interpolate in the standard normal table justuse the value in the table closest to the value you need to do your problem.

No clarification questions may be asked during the exam.

Good luck!

(1) You are considering a method by which to calculate , an estimator of . To see howgood this method is, you simulate the estimator four times, yielding the following simulatedvalues of the estimator: 8, 12, 15, 13. The true value of is 10. Presuming that each

of the above simulated values is equally likely, determine the bias of as an estimator of

(a) bias 0.5

(b) 0.5 < bias1.5

(c) 1.5 < bias2.5

(d) 2.5 < bias3.5

(e) 3.5 < bias

8/13/2019 Old Math 478 Exam 2

2/14

2

(2) Six trials of an estimator of have been simulated: 9, 8, 10, X4, 5, 8. The true valueof is 8. If the MSE (mean squared error) of the estimator is 5, and the bias of the estimatoris greater than zero, what is the bias of the estimator of ?

(a) bias 0.5

(b) 0.5 < bias1.5

(c) 1.5 < bias2.5

(d) 2.5 < bias3.5

(e) 3.5 < bias

(3) The random variableX has an exponential distribution with mean . Consider an estimator

of the mean, X

k

k

1

. Assuming that k= 9, determine the mean squared error of this

estimator of .

(a) 201.0

(b) 218.0

(c) 264.0

(d) 282.0

(e) None of answers (a) through (d) is correct.

(4) From a population having probability density functionfand a distribution function F, you aregiven the following sample:

8, 10, 10, 11, 11, 11, 12, 13, 13, 15

Use the uniform kernel with a bandwidth of 2.5 to estimate the probability, P, of observing avalue from this population greater than 12.

(a) P 0.25

(b) 0.25 < P0.30

(c) 0.30 < P0.35

(d) 0.35 < P0.45

(e) 0.40 < P

8/13/2019 Old Math 478 Exam 2

3/14

3

(5) Suppose each observation, iX , in a sample comes from a common distribution with mean

and variance ),( 2 . Which of the following statements isfalse?

(a) The variance of the sample mean X is n2 .

(b) The estimator

n

j

jn XXn

S1

22 )(1 is a biased estimator of the variance 2 .

(c) For a given exponential distribution, the method of moments and the maximumlikelihood estimators of the mean give the same value.

(d) For an unbiased estimator, the mean squared error of the estimator is equal to thevariance of the estimator.

(e) The estimator

n

j

jn XX

n

S

1

221 )(

1

1is a biased estimator of the variance 2 .

(6) You are given the following data:

1, 1, 2, 2, 4, 6, 6, 7, 9, 11, 12, 15, 17

Find the smoothed empirical estimate, , of the 70thpercentile.

(a) 8.5

(b) 8.5 < 9.0

(c) 9.0 < 9.5

(d) 9.5 < 10

(e) 10 <

8/13/2019 Old Math 478 Exam 2

4/14

4

(7) In a sample of size ten, you observe the following values:

3, 6, 1, 5, 8, 6, 1, 1, 3, 6

Determine the Nelson-alen estimate of the cumulative hazard rate function atx= 5, i.e.,

)5(H .

(a) )5(H 0.65

(b) 0.65 < )5(H 0.75

(c) 0.75 < )5(H 0.85

(d) 0.85 < )5(H 0.95

(e) 0.95 < )5(H

(8) You are given the following exposure and loss data from a survival study (where yjis thetime of death for each of a cohort of 20 people):

j yj s r

1 2 7 202 3 5 133 5 4 84 8 3 45 12 1 1

Using the Kaplan-Meier estimate of the cumulative hazard rate at time yj= 8, determine an

estimate the survival function, )8(S .

(a) 0.05

(b) 0.10

(c) 0.20

(d) 0.40

(e) 0.50

8/13/2019 Old Math 478 Exam 2

5/14

5

Use the following information and charts to answer problems (9) and (10) .

You are given the following grouped data, showing the number loss payments in each size-of-lossinterval:

Payment Range (x) Number of Payments0 < x 10 40

10 < x 20 2520 < x 30 1530 < x 50 1050 < x 100 10

Chart A

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

x

F(x)

Chart B

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

x

F(x)

Chart C

0

0.01

0.02

0.03

0.04

0.05

0 20 40 60 80 100

x

f(x)

Chart D

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100

x

f(x)

Chart E

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

x

f(x)

8/13/2019 Old Math 478 Exam 2

6/14

6

(9) Which of the charts above represents a correct ogive for this grouped data?

(a) Chart A.

(b) Chart B.

(c) Chart C.

(d) Chart D.

(e) Chart E.

(10) Which of the charts above represents a correct histogram for this grouped data?

(a) Chart A.

(b) Chart B.

(c) Chart C.

(d) Chart D.

(e) Chart E.

(11) The observations 5, 10, 25, 50, and 250 were obtained as a random sample from a Gammadistribution with unknown parameters and . Determine the method of moments estimateof .

(a) 120

(b) 120 < 130

(c) 130 < 140

(d) 140 < 150

(e) 150 <

8/13/2019 Old Math 478 Exam 2

7/14

7

(12) A random variable is suspected of being a distribution with probability density functionf(x)=exp(-x), x0. You observe the following sample of 11 values from the distribution:1, 1, 2, 3, 3, 4, 5, 6, 8, 10, and 12. Determine the estimate of using percentilematching on the median.

(a) 0.10

(b) 0.12

(c) 0.15

(d) 0.17

(e) 0.20

(13) Consider a pdf of the form 1)( ccxxf , 0 x 1. Let n= 40, and let the maximum

likelihood estimate c = 2. Determine the asymptotic variance of this MLE.

(a) 0.10

(b) 0.20

(c) 1.00

(d) 5.00

(e) 10.00

8/13/2019 Old Math 478 Exam 2

8/14

8

(14) N/A

8/13/2019 Old Math 478 Exam 2

9/14

9

(15) The observations 2, 5, 7, 14, and 20 were obtained as a random sample from a two-

parameter Pareto distribution with = 3, and unknown parameter . Estimate bypercentile matching, using the empirical smoothed estimate of the 75thpercentile. Then usethat result to estimate the limited expected value of this distribution at 10.

8/13/2019 Old Math 478 Exam 2

10/14

10

(16) You believe that a Poisson distribution, with a mean of , reflects the number of claims per

policy each year. You observe how many claims occur under each of three policies during ayear, and compile the following observation data:

Claims per Policy Number of Observations0 claims 2

1 or more claims 1

Calculate the maximum likelihood estimate of . Show your development of the likelihoodfunction.

8/13/2019 Old Math 478 Exam 2

11/14

11

(17) An insurance policy has a deductible of 20 and a policy limit (maximum payment per loss)of 80. The four loss payments under this policy have been 10, 30, 50, and 80. The insureris only aware of those losses on which a payment is made. Assuming the ground-up losses

were generated by a distribution of the form

x

exf

1

)( ,x> 0, find the maximum

likelihood estimate of . Show your development of the likelihood function.

8/13/2019 Old Math 478 Exam 2

12/14

12

Standard Normal Distribution Values

x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998

3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

8/13/2019 Old Math 478 Exam 2

13/14

13

Distribution Information

8/13/2019 Old Math 478 Exam 2

14/14

14