t2 - Stat 5102 Second Midterm Exam Name April 7 2010...

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Stat 5102 Second Midterm Exam April 7, 2010 Name Student ID The exam is closed book and closed notes. You may use one 8 1 2 × 11 sheet of paper with formulas, etc. You may also use the handouts on “brand name distributions” and Greek letters. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from. IID means independent and identically distributed; PDF means proba- bility density function; MLE means maximum likelihood estimate. The points for the questions total to 100. There are 6 pages and 5 prob- lems. 1. [20 pts.] Suppose X 1 , . . . , X n are IID N (0 , ν ), where ν is an unknown parameter satisfying ν > 0. (a) Find the log likelihood for ν . (b) Find the MLE for ν . 1
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(c) Show that your MLE is a local maximizer of the log likelihood. 2
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2. [20 pts.] Suppose X 1 , . . . , X n are IID having PDF f θ ( x ) = θ ( θ + 1) x θ - 1 (1 - x ) , 0 < x < 1 where θ > 0 is an unknown parameter.
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