411_hw01_soln

411_hw01_soln - Stat 411 Homework 01 Solutions 1. Since the...

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Stat 411 – Homework 01 Solutions 1. Since the X i ’s are independent, E ( Y n ) = E ( X 1 ) ··· E ( X n ) = c n . By Markov’s in- equality, P ( Y n > ε ) ε - 1 E ( Y n ) = ε - 1 c n 0 , n → ∞ , since c (0 , 1). Since ε > 0 is arbitrary, Y n p 0. 2. (a) The maximum of a list of numbers less than x if and only if all the numbers are less than x . Therefore, for any x (0 , 1), F M n ( x ) = P ( M n x ) = P ( X 1 x,. ..,X n x ) = P ( X 1 x ) n = x n ; if x 0, then F M n ( x ) = 0, and if x 1, then F M n ( x ) = 1. (b) Let Z n = n (1 - M n ). The CDF of Z n can be found as follows: F Z n ( z ) = P { n (1 - M n ) z } = P { M n 1 - z n } = 1 - (1 - z n ) n , z > 0 . A well-known fact from calculus about the exponential function gives lim n →∞ F Z n ( z ) = 1 - lim n →∞ (1 - z n ) n = 1 - e - z . This limit is the CDF of an exponential random variable Z with mean 1. 3. Write
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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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411_hw01_soln - Stat 411 Homework 01 Solutions 1. Since the...

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