HW1Soln - IEOR 161 - Introduction to Stochastic Processes...

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IEOR 161 - Introduction to Stochastic Processes Spring 2010 HW1 Solutions ** Note that the numbering is from Ross 9th Edition 2.45 Let N i denotes the number of keys in box i , i = 1 ,...,k . Then, with X equal to the number of collisions we have that, X = k X i =1 ( N i - 1) + = k X i =1 ( N i - 1 + I { N i = 0 } ) where I { N i = 0 } is an indicator function, which equals to 1 if N i = 0 and 0 otherwise. Now we have, E [ X ] = k X i =1 ( rp i - 1 + (1 - p i ) r ) = r - k + k X i =1 (1 - p i ) r 3.5 a) Let Z = number of red balls selected, we have for all i = 0 , 1 , 2 , 3 P ( X = i | Y = 3) = P ( X = i,Y = 3 ,Z = 3 - i ) P ( Y = 3) = ± 3 i ²± 5 3 ²± 6 3 - i ² ± 9 3 ²± 5 3 ² = ± 3 i ²± 6 3 - i ² ± 9 3 ² b) Given Y=1, we know that the other five balls are either white or red. For each ball selected, there is a 3 9 probability that it is a white ball. Hence, E [ X | Y = 1] = 5 × 3 9 = 5 3 3.7 Given Y = 2, we have P ( Y = 2) = 5 / 16. Hence, the conditional distribution of X and 1
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Z is P (( X,Z ) = (1 , 1) | Y = 2) = 1 / 16
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This note was uploaded on 02/10/2010 for the course IEOR 161 taught by Professor Lim during the Spring '08 term at Berkeley.

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HW1Soln - IEOR 161 - Introduction to Stochastic Processes...

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