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StFi2006Condensed - STAT 116 STANFORD UNIVERSITY Department...

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STAT 116 STANFORD UNIVERSITY Department of Statistics FINAL EXAMINATION STATISTICS 116 Instructor: Professor Anthony D’Aristotile August 19, 2006 3:30 pm - 6:30 pm IMPORTANT: Please, show all your work and justify your assertions. There are 200 total points. STUDENT Name: STUDENT ID : 1
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1. (20 points) Let X 1 , X 2 , ..... , X 100 be independent Poisson random variables with mean equal to 1. Use the central limit theorem to approximate P ( 100 1 X i > 75) 2. (20) From a signpost that says OHIO two letters fall off. A friendly drunk puts the two letters back into the two empty slots at random. What is the probability that the sign still says OHIO? 3. (7 + 7 + 6) Let X, X 1 , X 2 , ..., X n be identically distributed discrete random variables. (a) Find E ( X | X = k ) . (b) Find E ( X | X ) (c) Find E ( X 1 + X 2 | X 1 + X 2 + X 3 + X 4 + X 5 ) . 4. (5 + 10 + 10 + 5) Let X and Y be independent standard normal random variables. (a) Find the probability density function (p.d.f.) of ( X, Y ) . (b) Find the p.d.f. of ( U, V ) where U = Xcos θ - Y sin θ V = Xsin θ + Y cos θ (c) Find the p.d.f.’s of U and V .
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