HW11b

HW11b - n independent exponential random variables 3 Let X...

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1. Let X n Beta( n,n ), n =1 , 2 , 3 , · · · . Show that X n P (1 / 2) [i.e., show that P ( | X n - 1 / 2 | > ± ) 0 as n →∞ for every ± > 0. Use Chebyshevs’s inequality.] 2. Slutsky’s theorem states the following: if X n D X and Y n P a where a ± = 0 is a constant, then X n /Y n D X/a (here Z n D Z means that P ( Z n z ) P ( Z z ) for every z ( -∞ , ) at which the cdf of Z is continuous). Use this to prove that for X n Beta( n,n ) one has n ( X n - 1 / 2) D Normal(0 , σ 2 ). Find the value of σ 2 . [Hint: Use the fact that X n = W n / ( W n + V n ) where W n and V n are independent Gamma( n, 1) variables each of which can be written as the sum of
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Unformatted text preview: n independent exponential random variables!] 3. Let X 1 ,X 2 , ,X n Uniform(0 , ) where &amp;gt; 0. Can you give formulas for L n = L ( X 1 , ,X n ) and U n = U ( X 1 , ,X n ) such that P ( L n &amp;lt; &amp;lt; U n ) = 0 . 95. [Hint: Look at the distribution of M n = max( X 1 , ,X n ), the central limit theorem is of NO help here!]...
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This note was uploaded on 01/16/2011 for the course STAT 213 taught by Professor Ioannam during the Fall '09 term at Duke.

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