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hmwk1sol - ISyE 3232 Stochastic Manufacturing and Service...

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ISyE 3232 Stochastic Manufacturing and Service Systems Spring 2011 H.Ayhan Solutions to Homework 1 1. (a) Poisson (b) exponential (c) geometric (d) Bernoulli (e) binomial (f) normal 2. Since E [ X ] = 4 and Var( X ) = 100, ( a ) E [6 - 4 X ] = 6 - 4 E [ X ] = - 10 and Var(6 - 4 X ) = 16Var( X ) = 1 , 600. ( b ) E [( X - 3) / 5] = 1 5 E [ X ] - 3 5 = 1 / 5 and Var(( X - 3) / 5) = 1 25 Var( X ) = 4. 3. ( a ) 1 = 3 X k =1 P ( X = 2 k - 1) = 3 X k =1 (2 k - 1) c = 9 c so it follows that c = 1/9. ( b ) E [ X ] = 3 i =1 (2 k - 1) P ( X = 2 k - 1) = (1 / 9)(1 + 9 + 25) = 35 / 9 ( c ) E [ X 2 ] = 3 i =1 (2 k - 1) 2 P ( X = 2 k - 1) = (1 / 9)(1 + 27 + 125) = 17 ( d ) Var( X ) = E [ X 2 ] - ( E [ X ]) 2 = 17 - (35 / 9) 2 = 1 . 8765 ( e ) Note that ( X - 2) + = 0 with probability 1 / 9 1 with probability 3 / 9 3 with probability 5 / 9 Hence, E [( X - 2) + ] = 0 · (1 / 9) + 3 / 9 + 3 × 5 / 9 = 2 4. ( a ) p k = P ( X = k ) = e - 5 5 k k ! , k
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hmwk1sol - ISyE 3232 Stochastic Manufacturing and Service...

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