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Second Midterm 2000 - Second Midterm AEB 6933 1 Derive the...

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Second Midterm AEB 6933 1. Derive the maximum likelihood estimator for λ in the exponential distribution ( ) x e x f λ λ λ 1 1 = (This derivation should include explicit derivation of the likelihood function). Using the data in column 1 of the data table, estimate λ . 2. Using the results from question (1) test for the hypothesis that 8< λ<9. 3. Derive the method of moments estimator for λ for the exponential distribution in equation (1). 4. What is the asymptotic distribution of the maximum likelihood estimate of λ in question (1)? How do you justify this distribution? 5. Using column 2 in the data table and assuming that the random variable is normally distributed, what is the .05 confidence interval around the sample average? What is the difference between assuming that the variance is known versus unknown?
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Table 1. Test Data (1) (2) (3)=(2) 2 1 11.3850 9.2765 86.0535 2 10.0422 7.1405 50.9868 3 7.2711 9.4056 88.4652 4 25.8137 8.1577 66.5485 5 0.9803 15.1388 229.1827 6 12.3170 10.6836 114.1396 7 0.7303 9.4561 89.4171 8 0.6686 2.9272 8.5687 9 4.5317 9.1033 82.8697 10 4.5390 11.5320 132.9871 11 31.5965 12.9578 167.9039 12 12.6260 6.8197 46.5077 13 3.0875 5.0612 25.6154 14 2.9506 7.4032 54.8070 15 6.2673 9.6594 93.3045 16 23.4469 12.5749 158.1275 17 16.3294 8.2525 68.1031 18 9.7973 13.0351 169.9138 19 25.0235 3.0948 9.5775 20 15.2855 8.1196 65.9276 21 36.4097 12.1564 147.7781 22 1.3511 13.1377 172.5994 23 16.3487 15.4851 239.7897 24 7.3497 12.7546 162.6795 25 3.4824 17.5617 308.4136 26 6.7868
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