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Unformatted text preview: L 2 iF and only iF n =1 b 2 n < . 4. In this problem U 1 , U 2 , . . ., are independent and identically distributed random vari-ables, each uniFormly distributed on the inteval [0 , 1]. (a) Compute E [log U k ]. (b) Defne T n := ( U 1 U 2 U n ) 1 /n . Show that T n P e-1 . [Hint: Take logs, use (a) and the WLLN.] 5. Let X 1 , X 2 , . . . be independent random variables, with X n 0 For each n . Show that n X n < a.s. iF and only iF n E [min( X n , 1)] < ....
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- Fall '08