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Unformatted text preview: IE 336: Operations Research  Stochastic Models Homework Set #2 Solutions Spring 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 ) = parenleftbigg n 1 parenrightbigg p (1 p ) n 1 + parenleftbigg n 2 parenrightbigg p 2 (1 p ) n 2 + parenleftbigg n 3 parenrightbigg p 3 (1 p ) n 3 (c) E ( x ) = nq = 1 4 n 2. We need the parameters, a and b, to evaluate the uniform distribution. These can be obtained due to knowledge of the expectation and variance (i.e. via simultaneous equations) 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 ] = integraltext x x n f ( x ) dx = integraltext b a x n 1 b a dx = 1 b a bracketleftBig 1 n +1 x n +1 bracketrightBig b a = 1 b a bracketleftBig b n +1 a n +1 n +1 bracketrightBig 10 = 1 3 1 bracketleftBig 3 n +1 1...
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 Spring '08
 Leyla,O
 Operations Research, Probability theory, independent random variables, stochastic models, Binomial Probability Mass, xn b−a dx

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