EE465-USC-Fall08-Assignment2_solution

EE465-USC-Fall08-Assignment2_solution - EE 465 : Homework 2...

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Unformatted text preview: EE 465 : Homework 2 Solution 1. Each flip after the first will independently result in a changeover with probability 1 2 . Thus, we have a changeover possibility in n- 1 locations, implying: P ( k changeovers ) = n- 1 k 1 2 k 1 2 n- 1- k = n- 1 k 1 2 n- 1 2. a) X= number of red balls removed before the first black ball is chosen X i = n 1 , if redball iistakenbeforeany black ball , otherwise X = number of timesthatX i = 1 X = n X i =1 X i b) E [ X ] = E [ n X i =1 X i ] = n X i =1 E [ X i ] E [ X i ] = 1 X j =0 jP ( X i = j ) = 0 P ( X i = 0) + 1 P ( X i = 1) = P ( X i = 1) P ( X i = 1) = P ( Redball iischosenbeforeall mblack balls ) = 1 m + 1 since the i th red ball and the m black balls are equally likely to be the one chosen earliest. Therefore, E [ X ] = n i =1 E [ X i ] = n i =1 1 m +1 = n m +1 3. a) X i ( t i ) = (0 , ,..., , 1 , ,..., 0) with the 1 at the i th place b) ( t 1 ,t 2 ,...,t n ) = E ( e t 1 X 1 + t 2 X 2 + ... +...
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EE465-USC-Fall08-Assignment2_solution - EE 465 : Homework 2...

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