Management and Systems Engineering

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Department of Engineering Management and Systems Engineering Management
School of Engineering and Applied Science
The George Washington University
Examination IV Part II
EMSE 4770
Directions: Answer every question. For full credit, clearly show ALL work and
indicate how the solution was determined. The work must be neat and done on the
exam sheet or I will not grade it. The exam should reflect you own work. Rules of
Academic Integrity will be strictly enforced. If you have questions, email me. The
exam must be mailed to me by Friday at 11:59pm. Note there are helpful tables at
the back of the exam.
1. (20 points). The time to failure for a particular component is given by
F(x) = 1 – e-(x)2
where X is in months. Experts claim the following prior distribution for .
a. In a 1 month test of 5 components, 2 failed before the test ended (i.e. before 1
month), 1 component failed at time t=.5 and the other at t=.8, and 2 were still
functioning when the test was terminated. What is the posterior distribution
for .
i i Pr(=i|Data)
1 2 3
b. What is the posterior estimate for ?
c. What is the posterior estimate for the reliability at time t=1?
d. What is the posterior estimate for the mean failure time for the component?

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2. (20 points). In conducting and Expert Judgement exercise, you obtain the
following expert inputs from 3 experts for 4 seed variable
Exp 1 Exp 2 Exp 3
Seed Realisation 5% 50% 95% 5% 50% 95% 5% 50% 95%
1 11 20 30 40 15 25 50 20 40 60
2 25 10 20 30 10 25 50 40 45 50
3 15 10 30 50 10 50 100 10 40 50
4 65 40 50 60 20 40 80 10 20 80
a. Who is the most calibrated expert (you will have to approximate this given
the tables attached)?
b. Who is the most Informative Expert?
c. Now experts are asked to assess an unknown quantity and their
assessments are
Exp 1 Exp 2 Exp 3
5% 50% 95% 5% 50% 95% 5% 50% 95%
50 70 90 20 50 90 40 70 100
What is your best estimate of this unknown quantity?
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TABLES
x G(x) z F(z)
0.05 19.4701 -3.00 0.001
0.10 9.5135 -2.75 0.003
0.15 6.2203 -2.50 0.006
0.20 4.5908 -2.25 0.012
0.25 3.6256 -2.00 0.023
0.30 2.9916 -1.75 0.04
0.35 2.5461 -1.50 0.067
0.40 2.2182 -1.25 0.106
0.45 1.9681 -1.00 0.159
0.50 1.7725 -0.75 0.227
0.55 1.6161 -0.50 0.309
0.60 1.4892 -0.25 0.401
0.65 1.3848 0.00 0.5
0.70 1.2981 0.25 0.599
0.75 1.2254 0.50 0.691
0.80 1.1642 0.75 0.773
0.85 1.1125 1.00 0.841
0.90 1.0686 1.25 0.894
0.95 1.0315 1.50 0.933
1.00 1.0000 1.75 0.96
2.00 0.977
2.25 0.988
2.50 0.994
2.75 0.997
3.00 0.999
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Cdf of Chi Square Dist Cdf of Chi Square Dist
degrees of freedom degrees of freedom
x 2 3 4 5 6 7 x 2 3 4 5 6 7
0.1 0.0488 0.0082 0.0012 0.0002 0.0000 0.0000 8 0.9817 0.9540 0.9084 0.8438 0.7619 0.6674
0.2 0.0952 0.0224 0.0047 0.0009 0.0002 0.0000 8.1 0.9826 0.9560 0.9120 0.8492 0.7691 0.6761
0.3 0.1393 0.0400 0.0102 0.0024 0.0005 0.0001 8.2 0.9834 0.9579 0.9155 0.8544 0.7762 0.6847
0.4 0.1813 0.0598 0.0175 0.0047 0.0011 0.0003 8.3 0.9842 0.9598 0.9188 0.8595 0.7831 0.6931
0.5 0.2212 0.0811 0.0265 0.0079 0.0022 0.0006 8.4 0.9850 0.9616 0.9220 0.8645 0.7898 0.7014
0.6 0.2592 0.1036 0.0369 0.0120 0.0036 0.0010 8.5 0.9857 0.9633 0.9251 0.8693 0.7963 0.7094
0.7 0.2953 0.1268 0.0487 0.0170 0.0055 0.0017 8.6 0.9864 0.9649 0.9281 0.8739 0.8026 0.7173
0.8 0.3297 0.1505 0.0616 0.0230 0.0079 0.0026 8.7 0.9871 0.9664 0.9309 0.8784 0.8088 0.7251
0.9 0.3624 0.1746 0.0754 0.0298 0.0109 0.0037 8.8 0.9877 0.9679 0.9337 0.8827 0.8149 0.7327
1 0.3935 0.1987 0.0902 0.0374 0.0144 0.0052 8.9 0.9883 0.9693 0.9364 0.8869 0.8207 0.7401
1.1 0.4231 0.2229 0.1057 0.0459 0.0185 0.0069 9 0.9889 0.9707 0.9389 0.8909 0.8264 0.7473
1.2 0.4512 0.2470 0.1219 0.0551 0.0231 0.0091 9.1 0.9894 0.9720 0.9414 0.8949 0.8320 0.7544
1.3 0.4780 0.2709 0.1386 0.0651 0.0283 0.0116 9.2 0.9899 0.9733 0.9437 0.8987 0.8374 0.7614
1.4 0.5034 0.2945 0.1558 0.0757 0.0341 0.0144 9.3 0.9904 0.9744 0.9460 0.9023 0.8426 0.7682
1.5 0.5276 0.3177 0.1734 0.0869 0.0405 0.0177 9.4 0.9909 0.9756 0.9482 0.9059 0.8477 0.7748
1.6 0.5507 0.3406 0.1912 0.0988 0.0474 0.0214 9.5 0.9913 0.9767 0.9503 0.9093 0.8527 0.7813
1.7 0.5726 0.3631 0.2093 0.1111 0.0549 0.0254 9.6 0.9918 0.9777 0.9523 0.9126 0.8575 0.7876
1.8 0.5934 0.3851 0.2275 0.1239 0.0629 0.0299 9.7 0.9922 0.9787 0.9542 0.9158 0.8621 0.7938
1.9 0.6133 0.4066 0.2459 0.1372 0.0713 0.0348 9.8 0.9926 0.9797 0.9561 0.9189 0.8667 0.7998
2 0.6321 0.4276 0.2642 0.1509 0.0803 0.0402 9.9 0.9929 0.9806 0.9579 0.9219 0.8711 0.8057
2.1 0.6501 0.4481 0.2826 0.1649 0.0897 0.0459 10 0.9933 0.9814 0.9596 0.9248 0.8753 0.8114
2.2 0.6671 0.4681 0.3010 0.1792 0.0996 0.0521 10.1 0.9936 0.9823 0.9612 0.9275 0.8795 0.8170
2.3 0.6834 0.4875 0.3192 0.1937 0.1099 0.0586 10.2 0.9939 0.9831 0.9628 0.9302 0.8835 0.8225
2.4 0.6988 0.5064 0.3374 0.2085 0.1205 0.0656 10.3 0.9942 0.9838 0.9643 0.9328 0.8874 0.8278
2.5 0.7135 0.5247 0.3554 0.2235 0.1315 0.0729 10.4 0.9945 0.9845 0.9658 0.9353 0.8912 0.8330
2.6 0.7275 0.5425 0.3732 0.2386 0.1429 0.0806 10.5 0.9948 0.9852 0.9672 0.9378 0.8949 0.8380
2.7 0.7408 0.5598 0.3908 0.2539 0.1546 0.0887 10.6 0.9950 0.9859 0.9686 0.9401 0.8984 0.8430
2.8 0.7534 0.5765 0.4082 0.2692 0.1665 0.0971 10.7 0.9953 0.9865 0.9698 0.9423 0.9019 0.8477
2.9 0.7654 0.5927 0.4253 0.2846 0.1787 0.1059 10.8 0.9955 0.9871 0.9711 0.9445 0.9052 0.8524
3 0.7769 0.6084 0.4422 0.3000 0.1912 0.1150 10.9 0.9957 0.9877 0.9723 0.9466 0.9085 0.8570
3.1 0.7878 0.6235 0.4588 0.3154 0.2038 0.1244 11 0.9959 0.9883 0.9734 0.9486 0.9116 0.8614
3.2 0.7981 0.6382 0.4751 0.3308 0.2166 0.1341 11.1 0.9961 0.9888 0.9745 0.9506 0.9147 0.8657
3.3 0.8080 0.6524 0.4911 0.3462 0.2296 0.1441 11.2 0.9963 0.9893 0.9756 0.9524 0.9176 0.8699
3.4 0.8173 0.6660 0.5068 0.3614 0.2428 0.1543 11.3 0.9965 0.9898 0.9766 0.9543 0.9205 0.8739
3.5 0.8262 0.6792 0.5221 0.3766 0.2560 0.1648 11.4 0.9967 0.9903 0.9776 0.9560 0.9232 0.8779
3.6 0.8347 0.6920 0.5372 0.3917 0.2694 0.1755 11.5 0.9968 0.9907 0.9785 0.9577 0.9259 0.8818
3.7 0.8428 0.7043 0.5519 0.4066 0.2828 0.1864 11.6 0.9970 0.9911 0.9794 0.9593 0.9285 0.8855
3.8 0.8504 0.7161 0.5663 0.4214 0.2963 0.1975 11.7 0.9971 0.9915 0.9803 0.9609 0.9310 0.8891
3.9 0.8577 0.7275 0.5803 0.4361 0.3098 0.2088 11.8 0.9973 0.9919 0.9811 0.9624 0.9334 0.8927
4 0.8647 0.7385 0.5940 0.4506 0.3233 0.2202 11.9 0.9974 0.9923 0.9819 0.9638 0.9358 0.8961
4.1 0.8713 0.7491 0.6074 0.4649 0.3369 0.2318 12 0.9975 0.9926 0.9826 0.9652 0.9380 0.8994
4.2 0.8775 0.7593 0.6204 0.4790 0.3504 0.2435 12.1 0.9976 0.9930 0.9834 0.9666 0.9402 0.9027
4.3 0.8835 0.7692 0.6331 0.4929 0.3639 0.2554 12.2 0.9978 0.9933 0.9841 0.9679 0.9423 0.9058
4.4 0.8892 0.7786 0.6454 0.5066 0.3773 0.2673 12.3 0.9979 0.9936 0.9847 0.9691 0.9444 0.9089
4.5 0.8946 0.7877 0.6575 0.5201 0.3907 0.2793 12.4 0.9980 0.9939 0.9854 0.9703 0.9464 0.9119
4.6 0.8997 0.7965 0.6691 0.5334 0.4040 0.2914 12.5 0.9981 0.9941 0.9860 0.9715 0.9483 0.9147
4.7 0.9046 0.8049 0.6805 0.5464 0.4172 0.3035 12.6 0.9982 0.9944 0.9866 0.9726 0.9502 0.9175
4.8 0.9093 0.8130 0.6916 0.5592 0.4303 0.3156 12.7 0.9983 0.9947 0.9872 0.9736 0.9519 0.9202
4.9 0.9137 0.8207 0.7023 0.5718 0.4433 0.3278 12.8 0.9983 0.9949 0.9877 0.9747 0.9537 0.9229
5 0.9179 0.8282 0.7127 0.5841 0.4562 0.3400 12.9 0.9984 0.9951 0.9882 0.9757 0.9553 0.9254
5.1 0.9219 0.8354 0.7228 0.5962 0.4689 0.3522 13 0.9985 0.9954 0.9887 0.9766 0.9570 0.9279
5.2 0.9257 0.8423 0.7326 0.6080 0.4816 0.3644 13.1 0.9986 0.9956 0.9892 0.9775 0.9585 0.9303
5.3 0.9293 0.8489 0.7421 0.6196 0.4940 0.3766 13.2 0.9986 0.9958 0.9897 0.9784 0.9600 0.9326
5.4 0.9328 0.8553 0.7513 0.6310 0.5064 0.3887 13.3 0.9987 0.9960 0.9901 0.9793 0.9615 0.9349
5.5 0.9361 0.8614 0.7603 0.6421 0.5185 0.4008 13.4 0.9988 0.9962 0.9905 0.9801 0.9629 0.9371
5.6 0.9392 0.8672 0.7689 0.6529 0.5305 0.4128 13.5 0.9988 0.9963 0.9909 0.9809 0.9643 0.9392
5.7 0.9422 0.8728 0.7773 0.6635 0.5424 0.4248 13.6 0.9989 0.9965 0.9913 0.9816 0.9656 0.9412
5.8 0.9450 0.8782 0.7854 0.6738 0.5540 0.4367 13.7 0.9989 0.9967 0.9917 0.9824 0.9668 0.9432
5.9 0.9477 0.8834 0.7933 0.6839 0.5655 0.4485 13.8 0.9990 0.9968 0.9920 0.9831 0.9680 0.9451
6 0.9502 0.8884 0.8009 0.6938 0.5768 0.4603 13.9 0.9990 0.9970 0.9924 0.9837 0.9692 0.9470
6.1 0.9526 0.8932 0.8082 0.7034 0.5879 0.4719 14 0.9991 0.9971 0.9927 0.9844 0.9704 0.9488
6.2 0.9550 0.8977 0.8153 0.7128 0.5988 0.4834 14.1 0.9991 0.9972 0.9930 0.9850 0.9715 0.9506
6.3 0.9571 0.9021 0.8222 0.7219 0.6096 0.4948 14.2 0.9992 0.9974 0.9933 0.9856 0.9725 0.9523
6.4 0.9592 0.9063 0.8288 0.7308 0.6201 0.5061 14.3 0.9992 0.9975 0.9936 0.9862 0.9735 0.9539
6.5 0.9612 0.9103 0.8352 0.7394 0.6304 0.5173 14.4 0.9993 0.9976 0.9939 0.9867 0.9745 0.9555
6.6 0.9631 0.9142 0.8414 0.7479 0.6406 0.5283 14.5 0.9993 0.9977 0.9941 0.9873 0.9755 0.9570
6.7 0.9649 0.9179 0.8474 0.7561 0.6505 0.5392 14.6 0.9993 0.9978 0.9944 0.9878 0.9764 0.9585
6.8 0.9666 0.9214 0.8532 0.7641 0.6603 0.5500 14.7 0.9994 0.9979 0.9946 0.9883 0.9773 0.9600
6.9 0.9683 0.9248 0.8587 0.7718 0.6698 0.5606 14.8 0.9994 0.9980 0.9949 0.9887 0.9781 0.9613
7 0.9698 0.9281 0.8641 0.7794 0.6792 0.5711 14.9 0.9994 0.9981 0.9951 0.9892 0.9790 0.9627
7.1 0.9713 0.9312 0.8693 0.7867 0.6883 0.5815 15 0.9994 0.9982 0.9953 0.9896 0.9797 0.9640
7.2 0.9727 0.9342 0.8743 0.7938 0.6973 0.5916 15.1 0.9995 0.9983 0.9955 0.9901 0.9805 0.9653
7.3 0.9740 0.9371 0.8791 0.8007 0.7060 0.6017 15.2 0.9995 0.9983 0.9957 0.9905 0.9812 0.9665
7.4 0.9753 0.9398 0.8838 0.8074 0.7146 0.6115 15.3 0.9995 0.9984 0.9959 0.9908 0.9820 0.9677
7.5 0.9765 0.9424 0.8883 0.8140 0.7229 0.6213 15.4 0.9995 0.9985 0.9961 0.9912 0.9826 0.9688
7.6 0.9776 0.9450 0.8926 0.8203 0.7311 0.6308 15.5 0.9996 0.9986 0.9962 0.9916 0.9833 0.9699
7.7 0.9787 0.9474 0.8968 0.8264 0.7391 0.6402 15.6 0.9996 0.9986 0.9964 0.9919 0.9839 0.9710
7.8 0.9798 0.9497 0.9008 0.8324 0.7469 0.6494 15.7 0.9996 0.9987 0.9966 0.9922 0.9845 0.9720
7.9 0.9807 0.9519 0.9047 0.8382 0.7545 0.6585 15.8 0.9996 0.9988 0.9967 0.9926 0.9851 0.9730

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