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Unformatted text preview: j + 1)st toss is diFerent from the outcome of the j th toss. The number R of distinct runs is R = 1 + ∑ n1 j =1 I j . Observe that I j and I k are independent if  jk  > 1. Show that E [( R1) 2 ] = ( n1) E [ I 1 ] + 2( n2) E [ I 1 I 2 ] + (( n1) 2( n1)2( n2)) E [ I 1 ] 2 . Show that E [ I 1 I 2 ] = p 2 q + pq 2 = pq. 6. Ex. 3.4.2. Hint: Let T = ∑ k i =1 X i , where X i is the number on the i th ball. Show that: E [ T ] = 1 2 k ( n + 1). show that E [ T 2 ] = 1 6 k ( n + 1)(2 n + 1) + 1 12 k ( k1)(3 n + 2)( n + 1). Hint: N X k =1 k = n ( n + 1) 2 n X k =1 k 2 = n ( n + 1)(2 n + 1) 6 . n X k =1 k 3 = ± n ( n + 1) 2 ² 2 . 7. Ex 3.5.2. Hint: P ( H = x ) = ∑ ∞ n = x P ( H = x  N = n ) P ( N = n ). 8. Ex 3.6.5. Hint: log y ≤ y1, with equality if and only if y = 1. 2...
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 Spring '08
 Stites,M

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