Assignment 1 - 1/2010 THE UNIVERSITY OF HONG KONG...

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1/2010 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 M n = min { X 1 , . . . , X n } . (a) Show that P ( M n > x/n ) = 1 - x n · n for x [0 , n ]. (b) Deduce from (a) that nM n converges in distribution to X , where X has the exponential 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 be
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