# soln1 - 1 ISyE 6739 Test#2 Solutions Summer 2005 This test...

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1 ISyE 6739 — Test #2 Solutions Summer 2005 This test is open notes, open books. You have exactly 90 minutes . Just do the best you can, and good luck! 1. (30 points) Short Answer Questions. (a) Suppose X has p.d.f. f ( x ) = 5 x 4 , 0 x 1. Find E [2 X - 5]. ANSWER: 2 E [ X ] - 5 = - 10 / 3. (b) If X again has p.d.f. f ( x ) = 5 x 4 , 0 x 1, find Var (2 X - 5). ANSWER: 4 Var ( X ) = 0 . 0793. (c) Suppose X can equal 1 or 2, each with probability 1 / 2. Find E [ n( X )]. ANSWER: 1 2 n(2) = 0 . 347. (d) TRUE or FALSE? E [ X 2 ] ( E [ X ]) 2 . ANSWER: True. (e) Suppose X has m.g.f. M X ( t ) = 0 . 3 e t + 0 . 7. What’s the distribution of X ? ANSWER: Bern(0.3). (f) If X has m.g.f. 4 / (4 - t ), for t < 4, find the m.g.f. of 2 X - 1. ANSWER: 4 e - t / (4 - 2 t ), t < 2. (g) TRUE or FALSE? If Cov ( X, Y ) = 0, then X and Y are independent. ANSWER: False.

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2 (h) Suppose that X and Y are independent Exponential( λ ) random variables. What is the m.g.f. of X + Y ? ANSWER: ( λ/ ( λ - t )) 2 . (i) Suppose that X and Y are independent Exponential( λ ) random variables. Find Var ( XY ) (yup — the variance of the product).
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• Spring '10
• tricahuu
• Probability theory, var, short answer questions., Joey Yi

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