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hw2 - solutions - f08

hw2 - solutions - f08 - IE 336 Operations Research...

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IE 336: Operations Research - Stochastic Models Homework Set #2 Solutions Fall 2008 1. (a) Let p = P ( Chip = Good ) = 3 4 , and let q = P ( Chip = Bad ) . Then, q = 1 - P ( Chip = Good ) = 1 - p = 1 - 3 4 = 1 4 (b) Let X = Number of bad chips in group of n X Binomial ( n, p ) Then, P ( X x ) = n 0 p 0 (1 - p ) n + n 1 p (1 - p ) n - 1 + n 2 p 2 (1 - p ) n - 2 + n 3 p 3 (1 - p ) n - 3 (c) Since ”the number of bad chips” follows binomial distribution, E ( x ) = nq = 1 4 n 2. The mean and variance of the uniform distribution U(a,b) are, E ( X ) = a + b 2 , V ar ( X ) = ( b - a ) 2 12 . So, E ( X ) = a + b 2 = 2 and, V ar ( X ) = ( b - a ) 2 12 = 1 3 . Solving for a and b we obtain a = 1 and b = 3. E [ X n ] = R x x n f ( x ) dx = R b a x n 1 b - a dx = 1 b - a h 1 n +1 x n +1 i b a = 1 b - a h b n +1 - a n +1 n +1 i 10 = 1 3 - 1 h 3 n +1 - 1 n +1 i 20 = 3 n +1 - 1 n +1 20 n + 21 = 3 n +1 There is no easy way to solve for n, therefore we accept this as the answer. However, knowledge of any technique be it theoretical or practical (i.e. Goal Seek in Excel) will yield an answer of n = 3. 1
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3. We know the set of boxes is { y = Yellow, g = Green } and the set of colors is { r = Red, b = Blue } Let B = selected box and C = selected color.
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