# hw12 - var N n = n 2 n-1 X k =1 1/k 2-n n-1 X k =1 1/k ≤...

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MATH 471 HW 12 Solutions Pengsheng Ji December 7, 2006 5.1.2 Note EX = 4 · 1 / 2 = 2, var ( X ) = 4 · (1 / 2) 2 = 1. Chebyshev’s upper bound is P ( | X - 2 | ≥ 2) var ( X ) 2 2 = 1 4 . The exact probability is P ( | X - 2 | ≥ 2) = P ( X = 0)+ P ( X = 4) = (1 / 2) 4 +(1 / 2) 4 = 1 / 8. 5.1.10 (a)The deﬁnition of M n and the independence between X i ’s, P ( M n > ² ) = ( P ( X 1 > ² )) n = (1 - ² ) n 0, as n → ∞ . (b) P ( M n > x/n ) = (1 - x/n ) n e - x , as n → ∞ . 5.1.17 Chebyshev’s inequality gives P ±ﬂ N n - EN n nlnn ² var ( N n ) ² 2 n 2 ln 2 n . ( * ) Since, from previous homework, we have got
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Unformatted text preview: var ( N n ) = n 2 n-1 X k =1 1 /k 2-n n-1 X k =1 1 /k ≤ n 2 ∞ X k =1 1 /k 2 , the right-hand side of (*) converges to 0 and ( N n-EN n ) / ( nlnn ) converges to 0 in probability. On the other hand, we know E ( N n ) nln = ∑ n k =1 1 /k lnn → 1 , as n → ∞ . Therefore, N n / ( nln ) converges to 1 in probability....
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