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hw2 - solutions

# hw2 - solutions - IE 336 Operations Research Stochastic...

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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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