Second Midterm 1999 - between.25 and.60 in each case 5 How...

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Second Midterm AEB 6933 1. Using column 1 in the sample table below, test the hypothesis that 0 ≤θ≤ 1 versus 0> θ or θ >1. 2. The exponential distribution function can be expressed as: ( ) 0 , 1 = x e x f x β β β Design the maximum likelihood estimator of β and estimate this value using column 2 in the sample table below. Test the hypothesis that β =5 at a .05 confidence level. 3. Using column 3 in the sample table, what is the .05 confidence interval around the sample average? What is the difference between assuming that the variance is known versus unknown? 4. Assume that the prior distribution for a Bernoulli distribution is: ( ) 2 6 6 p p p f = The posterior distribution of p given a single observation x=0 or 1 is then: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 1 1 0 2 1 2 1 2 2 sin 6 6 1 4 6 6 1 6 6 1 x x x x x p p p p dp p p p p p p p p x p g x x x x x x + = = π π Evaluating the posterior using L’Hopitals rule yields: ( ) ( ) ( ) = = = 1 if 1 - p 12 - 0 if 1 12 2 2 x p x p p x p g What is the Bayesian estimator of p in each case? What is the probability that p lies
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Unformatted text preview: between .25 and .60 in each case? 5. How well does the normal distribution approximate the binomial distribution of 5 positive occurrences out of 25 draws given a probability of success of .4? 6. What is the method of moments estimator of β in question 2? Data for Test Question 1 Question 2 Question 3 Obs (1) (2) (3) 1 4.37046 1.20199 5.29497 2 0.24479 8.47799 2.29667 3 0.47682 0.88445 0.25094 4 4.46811 3.24800 4.77228 5 3.08131 0.81029-1.42873 6 -0.58275 3.67474 4.57343 7 0.67127 6.71023 2.14318 8 1.74480 10.61494 0.99078 9 -0.23411 11.39558 5.22586 10 1.17052 7.79335 2.97007 11 -2.13961 6.32205 7.07155 12 2.32355 6.10838 1.44718 13 -2.49499 4.75707 7.78969 14 0.89687 2.52394 5.38877 15 1.25450 13.42708 5.43025 16 1.36741 3.36613-1.14312 17 1.66377 11.93958 3.10425 18 1.47865 2.88261 1.02771 19 2.11535 1.71714 4.67523 20 1.24796 9.11149 3.43871...
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