hmwk1sol

# 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 ° k =1 P ( X =2 k 1) = 3 ° 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 =0 , 1 ,... ( b ) E ( X )=5 ( c ) Var( X )=5 ( d ) Since Y =m in( X, 3), Y can only take value of 0 , 1 , 2 , 3. Let q k ,k =1 , 2 , 3 be the p.m.f of Y .W ethu shav e , q 0 = P ( X = 0) = e 5 ,q 1 = P ( X = 1) = 5 e 5 q 2 = P ( X = 2) = 25 2 e 5 ,q 3 = P ( X 3) = ° i =3 e

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## This note was uploaded on 03/08/2012 for the course ISYE 3232 taught by Professor Billings during the Fall '07 term at Georgia Tech.

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hmwk1sol - ISyE 3232 Stochastic Manufacturing and Service...

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