HW1solns - February 2 2007 HW1 Problem Solutions#1.4.3...

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February 2, 2007 HW1 Problem Solutions #1.4.3. First, remark that by defnition the random variables V j = U j - U j - 1 are iid , where j 1 and U 0 0. These variables each represent the number o± trials up to an including the frst success, i.e. ±or k, m 1, P ( U k - U k - 1 = m ) = P ( V 1 = m ) = P ( Y 1 = Y 2 = ··· = Y m - 1 , Y m = 1) = (1 - p ) m - 1 p All o± the X k random variables are iid and independent o± the iid variables U j as well. We have to do three things: show that the variables S k are mutually independent, show that they all have the same distribution, and establish that the exact ±orm o± the distribution o± S 1 is exponenential as stated. For the frst and second parts: P ( S k > t | U k - 1 = n, X 1 , . . . , X n ) = P ( S k > t | U k - 1 = n, X 1 , . . . , X n ) = P ( V k X j =1 X n + j > t | U k - 1 = n ) = P ( V k X j =1 X j > t ) = P ( S 1 > t ) where the next-to-last step holds because X j are iid and X ’s are independent U ’s and V ’s.
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HW1solns - February 2 2007 HW1 Problem Solutions#1.4.3...

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