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# exam - Applied Statistics Comprehensive Examination...

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Unformatted text preview: Applied Statistics Comprehensive Examination In-Class Portion over STAT 519 and STAT 528 January 6, 2006, 1:00pm – 5:00pm This exam covers material in STAT 519 Probability Theory and STAT 528 Mathematical Statistics . You are allowed one sheet of formulas and a calculator. • Justify all of your answers and show all your work. An answer without justification is the same as no answer. • Be as clear and complete as possible. • Start each problem on a new page. • Write on only one side of each page. • Write your name on each page. • Do not staple your pages. • Do as many problems as you can. Notation: Define the following indicator function for any set A : I A ( x ) = ( 1 , if x ∈ A , if x / ∈ A 1. In a certain game a participant is allowed three attempts at scoring a hit. In the three attempts she must alternate which hand is used; thus she has two possible strategies: RLR (right, left, right) or LRL (left, right, left). The probabilities that she scores a hit with her right and left hands are p r and p l , respectively, and each attempt is independent of the others. Since she is right-handed, p r > p l . There are two versions of the game: I. She wins if she scores at least two hits. II. She wins if she scores at least two hits in a row. (a) Which strategy (RLR or LRL) maximizes her chance of winning version I of the game? (b) Which strategy (RLR or LRL) maximizes her chance of winning version II of the game? (c) If she uses the optimal strategy for each version of the game, which game (I or II) does she have the best chance of winning? 2. Let X 1 , X 2 , . . . , X n be a random sample from the density f ( x ; θ ) = θx- 2 I ( θ, ∞ ) ( x ) where θ > 0. (a) Find a maximum likelihood estimator of θ . (b) Find a sufficient statistic for θ . 3. Let X be a single observation from the density f ( x ; θ ) = ( θ/ 2) | x | (1- θ ) 1-| x | I {- 1 , , 1 } ( x ) where ≤ θ ≤ 1. (a) Is X a sufficient statistic for θ ? Is it complete? (b) Is | X | a sufficient statistic for θ ? Is it complete? (c) Does f ( x ; θ ) belong to the exponential family? (d) Is T = 2 I { 1 } ( x ) an unbiased estimator of θ . (e) Find a maximum likelihood estimator of θ ....
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exam - Applied Statistics Comprehensive Examination...

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