Solution let d be the demand we know that d exp ? ? 1

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Solution: Let D be the demand. We know that D Exp ( λ ) , λ = 1 1 . 8 = 0 . 56. The shortage can we written as: S ( D ) = ( D-1.5 if D 1 . 5 0 otherwise Then: E [ S ] = Z 1 . 5 ( x - 1 . 5) · 0 . 56 · e - 0 . 56 x dx = 0 . 78 E [ S 2 ] = Z 1 . 5 ( x - 1 . 5) 2 · 0 . 56 · e - 0 . 56 x dx = 2 . 8 V ar [ S ] = E [ S 2 ] - ( E [ S ]) 2 = 2 . 2 σ = 1 . 48
IEOR4101 - Probability Models - Assignment 6 - Veronica Miranda 2 Problem 3 In an Internet auction of a collector’s item ten bids are done. The bids are independent of each other and are uniformly distributed on (0, 1). The person with the largest bid gets the item for the price of the second largest bid. a) Show that the probability that the second largest bid exceeds the value x is equal to 10 k =2 ( 10 k ) (1 - x ) k · x 10 - k for x (0 , 1) b) Use the result from (a), and the identity: Z 1 0 x a (1 - x ) b = a ! b ! ( a + b + 1)! for any integers a, b 0, to obtain the expected value of the bid. Solution:

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