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hmwk2sol

# hmwk2sol - ISyE 3232 Stochastic Manufacturing and Service...

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ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2011 H. Ayhan Solutions to Homework 2 1. a) To calculate the pmf for min( D, 7), we should crete the following table, D 7 min( D, 7) P( D = x ) 5 7 5 1/10 6 7 6 2/10 7 7 7 2/5 8 7 7 2/10 9 7 7 1/10 From here, we can find the pmf, P(min( D, 7) = k ) = 1 / 10 if k = 5 2 / 10 if k = 6 7 / 10 if k = 7 0 otherwise Hence, E [min( D, 7)] = 7 d =5 d P(min( D, 7) = d ) = 1 10 5 + 2 10 6 + 7 10 7 = 6 . 6 b) By constructing the same type of table as in part (a.), we can construct the pmf of (7 D ) + by P ((7 D ) + = k ) = 1 / 10 if k = 2 2 / 10 if k = 1 7 / 10 if k = 0 0 otherwise Hence, E [(7 D ) + ] = 1 10 2 + 2 10 1 = 0 . 4 2. a) To assist calculating the expected value, we can construct the following table. D 8 max( D, 8) d 8 8 8 d > 8 8 d 1

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With the help of the table the expected value can be calculated. E [max( D, 8)] = 10 5 max( s, 8) d s = 8 5 8 5 d s + 10 8 s 5 d s = 8 P ( D 8) + 10 8 s 5 ds = 8 3 5 + s 2 10 10 8 = 24 5 + 36 10 = 8 . 4 b) This value can be calculated using a method similar to Problem 1.b., but first we need
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