Assignment1 - 1/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: 1/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 1 1. Let X 1 ,X 2 ,... be i.i.d. U [0 , 1] random variables. Define X ( n ) = max { X 1 ,...,X n } . (a) Show that P ‡ X ( n ) < 1- x n · = ‡ 1- x n · n for x ∈ [0 ,n ]. (b) Deduce from (a) that n ( 1- X ( n ) ) converges in distribution to X , where X has the expo- nential distribution of unit rate. 2. Let X 1 ,X 2 ,... be i.i.d. U [0 , 1] random variables. Define ¯ X n = n- 1 ∑ n i =1 X i . (a) Show, by the Weak Law of Large Numbers, that ¯ X converges in distribution to 1 / 2. (b) Show, by the Weak Law of Large Numbers, that n- 1 ∑ n i =1 X 2 i converges in distribution to 1 / 3. (c) Show, by the Central Limit Theorem, that √ n ( ¯ X n- 1 / 2) converges in distribution to a N (0 , 1 / 12) random variable. (d) The Central Limit Theorem asserts that n- 1 / 2 ∑ n i =1 ( X 2 i- 1 / 3) converges in distribution to a random variable Z . Specify the distribution of Z . 3. Let X 1 ,X 2 ,...,X 2 m be 2 m i.i.d. N (0 , 1) random variables, and X (1) ≤ X (2) ≤ ··· ≤ X (2 m ) be their sorted sequence. Denote by Φ the standard normal distribution function, which can betheir sorted sequence....
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.

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Assignment1 - 1/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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