hw8soln - . 1718 . Problem 4 Given that X Poi ( = 6), we...

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ORIE 3500/5500, Fall ’10 HW 8 Solutions Assignment 8 Solutions Problem 1 (a) Let X i be the demand of the i th week for i = 1 , 2, then X 1 ,X 2 ,i.i.d N (1000 , 200 2 ), and hence P ( X 1 < 1100 ,X 2 < 1100) = P ( X 1 < 1100) 2 = P ± X 1 - 1000 200 < 1100 - 1000 200 ² 2 = P ( N (0 , 1) < 0 . 5) 2 = 0 . 6915 2 = 0 . 4781 . (b) Note that X 1 + X 2 N (2000 , 2 × 200 2 ) by independence of X 1 and X 2 . So we get P ( X 1 + X 2 > 2200) = P ± X 1 + X 2 - 2000 2 × 200 2 > 2200 - 2000 2 × 200 2 ² = P ( N (0 , 1) > 1 / 2) = 0 . 23975 . Problem 2 (a) Let X denote the number of minutes that airline flights are behind the schedule. Since X Gamma (3 , 1 / 4), the pdf is given by f X ( x ) = (1 / 4) 3 Γ(3) x 3 - 1 e - x/ 4 = 1 128 x 2 e - x/ 4 , x > 0 . So the probability is given by P ( X > 15) = Z 15 1 128 x 2 e - x/ 4 dx = 1 128 ± 900 e - 15 / 4 + 8 Z 15 xe - x/ 4 dx ² by integration by parts = 1 128 ³ 900 e - 15 / 4 + 8 × 76 e - 15 / 4 ´ apply integration by parts to Z 15 xe - x/ 4 dx = 377 32 e - 15 / 4 = 0 . 2771 . (b) Since EX = α/λ = 12, P ( X > 15) EX 15 = 0 . 8 . This upper bound is about three times as large as the exact probability. 1
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ORIE 3500/5500, Fall ’10 HW 8 Solutions Problem 3 Since X χ 2 1 , Y χ 2 2 and X and Y are independent, X + Y χ 2 (1+2) = χ 2 3 . Using χ 2 3 table or statistical software, P [ X + Y > 5] = 1 - P [ X + Y 5] = 0
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Unformatted text preview: . 1718 . Problem 4 Given that X Poi ( = 6), we get P [ X 4] = 4 X i =0 P [ X = i ] = 4 X i =0 e-6 6 i i ! = 0 . 2851 . Using normal approximation, we get X- = X-6 6 N (0 , 1) . So, P [ X 4] = P X-6 6 -2 6 = -2 6 = 0 . 2071 . We know that the normal approximation is valid for large values of . So, even though here = 6 is not very large, the approximated probability is close (though certainly not very close) to the actual one. Problem 5 Let F ( ) be the cumulative distribution function of Y = 1 b X-a b . Then, F ( x ) = P 1 b X-a b x = P [ X bx + a ] = bx + a-a b = ( x ) . So, the distribution of Y is standard normal. So, X = bY + a N ( a,b 2 ) . 2...
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hw8soln - . 1718 . Problem 4 Given that X Poi ( = 6), we...

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