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SOLUTIONS%20TEST%202%20STA261

# SOLUTIONS%20TEST%202%20STA261 - TEST 2 STA261 c David...

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TEST 2 STA261 ( c David Brenner, 2010) Mar.24, 2010 name SOLUTIONS student number TA (1) Q1 Q2 Q3 Q4 total Instructions: No aids are allowed other than non-programmable calculators. Please show all your work clearly in the space provided to obtain partial credit ; you may use the back of the pages if necessary. TIME ALLOWED : 50 min. DO ANY 3 of the 4 QUESTIONS : approximate grade values are indicated.

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1. Suppose that X, Y IID G (3 , θ ) , θ > 0 a) Let T = X + Y and obtain both ET & varT as they depend on θ . W = T/ θ G (6) or T = θ W, W G (6) (5) ET = 6 θ & varT = 6 θ 2 b) Obtain a 95% confidence lower bound for the unknown value of θ . 2 T/ θ χ 2 (12) 95 / 100 = P (2 T/ θ 21 . 03) = P ( θ 2 T/ 21 . 03) (5) and therefore LB ( θ : . 95) = 2 T/ 21 . 03
2. Suppose that X 1 , . . . , X 8 IID N (0 , σ 2 ) , σ > 0 and let S = X 2 1 + · · · + X 2 8 a) Obtain both ES & varS as they depend on σ . S 2 / σ 2 χ 2 (8) or W = S 2 / 2 σ 2 G (4) (5) so S 2 = 2 σ 2 W, W G (4) ES = 2 σ EW 1 / 2 = 2 Γ (9 / 2) Γ (4) σ = 2 π · 35 32 σ varS = ES 2 - ( ES ) 2 = 8 - 2 π 35 32 2 σ 2 b) Obtain a 95% confidence lower

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