411_hw01 - Stat 411 – Homework 01 Due: Friday 01/20 1....

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Unformatted text preview: Stat 411 – Homework 01 Due: Friday 01/20 1. Let { X n : n ≥ 1 } be a sequence of positive independent random variables with E ( X n ) = c ∈ (0 , 1) for each n . Let Y n = X 1 X 2 ··· X n , the product of the X i ’s. Use Markov’s inequality to prove that Y n → 0 in probability. 2. Let X 1 ,...,X n be iid Unif (0 , 1) random variables. (a) Define M n = max { X 1 ,...,X n } . Find the CDF of M n . (b) Prove that n (1- M n ) → Z in distribution, where Z has CDF F Z ( z ) = 1- e- z . 3. Let X 1 ,...,X n be iid N ( μ, 1). Fix α ∈ (0 , 1) and define the random interval 1 C α ( X 1 ,...,X n ) = ( X- z ? 1- α/ 2 / √ n, X + z ? 1- α/ 2 / √ n ) , where X is the sample mean and z ? p is the 100 p th percentile of N (0 , 1), i.e., Φ( z ? p ) = p . Prove that C α ( X 1 ,...,X n ) has coverage probability 1- α . That is, prove P μ {C α ( X 1 ,...,X n ) contains μ } = 1- α....
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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