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SOLUTIONS%20FINAL%202007

# SOLUTIONS%20FINAL%202007 - SOLUTIONS 2007 STA261 c David...

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SOLUTIONS 2007 STA261 ( c David Brenner, 2007, 2010) revised Mar. 8, 2010

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1 2 3 4 5 7 8 9 10 total/70 UNIVERSITY OF TORONTO Faculty of Arts & Science APRIL-MAY EXAMINATIONS 2007 STA 261H1 S Prof. D. Brenner Duration - 3 hours Examination Aids: Non-programmable Calculators Instructions Please show all your work clearly in the space provided to obtain partial credit; you may use the backs of the pages for rough work. There are three parts to the exam: PARTS A, B and C You are to answer: 4 questions in part A, 1 in part B & 2 in part C for a total of 7 questions altogether. All complete questions will be valued equally, but partial grades are shown to the left of each part. Tables for the N (0 , 1) , χ 2 ( m ) , t ( m ) , & F ( m, n ) are appended. Name SOLUTIONS Student number TA (1)
PART A [Do any 4 questions of the 5 provided in this part] 1. Suppose that we have four competing probability functions { f θ | θ = 1 , 2 , 3 , 4 } for a certain random variable X distributed on X = { 1 , 2 , 3 , 4 , 5 } f 1 1 2 3 4 5 1 / 15 2 / 15 5 / 15 4 / 15 3 / 15 , f 2 1 2 3 4 5 1 / 15 4 / 15 3 / 15 2 / 15 5 / 15 f 3 1 2 3 4 5 6 / 15 2 / 15 2 / 15 2 / 15 3 / 15 , f 4 1 2 3 4 5 1 / 15 9 / 15 2 / 15 2 / 15 1 / 15 a) For θ = 2, what is the variance of X ? ( i.e. What is var θ X ?) (3) NA b) Determine the maximum likelihood estimator as a function θ : X -→ Θ . (2) NA c) For θ = 2, determine the distance of θ from θ : i.e. d ( θ , θ ) = E θ ( θ - θ ) 2 (5) NA

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2. Suppose that X N ( μ, σ 2 ) and that P ( X > 2 | X > 1) = P ( X > 1) = 2 P ( X > 2). a) Obtain μ , σ 2 and thus find P ( X > 3) P ( X > 2 | X > 1) = P ( X > 2) /P ( X > 1) = P ( X > 1) = 2 P ( X > 2) (6) P ( X > 2) = P ( X > 1) 2 = P ( X > 1) / 2 P ( X > 1) = 1 / 2 and P ( X > 2) = 1 / 4 P Z > 1 - μ σ = 1 2 and P Z > 2 - μ σ = 1 4 1 - μ σ = 0 and 2 - μ σ = . 67 μ = 1 and σ = 1 /. 67 P ( X > 3) = P Z > 3 - μ σ = P ( Z > 1 . 34) . 09 b) For what value of b is P ( X 2 / 2 b ) 1 / 100?
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SOLUTIONS%20FINAL%202007 - SOLUTIONS 2007 STA261 c David...

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