t309soln - STAT 330 Test 3 Solutions 1 Suppose X1 Xn is a...

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STAT 330 Test 3 Solutions 1 . Suppose X 1 ,...,X n is a random sample from the distribution with c.d.f. F ( x )= P ( X i x ( 1 1 x 2 if x 1 0 if x< 1 Find the limiting distribution of each of the following: ( a )( i )[2 ] Since P ¡ X (1) t ¢ =1 P ¡ X (1) >t ¢ n Q i =1 P ( X i )=1 n Q i =1 1 t 2 1 t 2 n ,t 1 (1) and P ¡ X (1) t ¢ =0 if t< 1 then lim n →∞ P ¡ X (1) t ¢ = ( lim n →∞ ¡ 1 1 t 2 n ¢ if t> 1 0 if 1 and therefore X (1) p 1 . ( ii )[3 ] Since P ( Y n y P £ n ( X (1) 1) y ¤ = P ³ X (1) 1+ y n ´ ³ y n ´ 2 n , if y 0 (2) by (1) and P ( Y n y )=0 if y< 0 then lim n →∞ P ( Y n y ( lim n →∞ h 1 ¡ y n ¢ 2 n i e 2 y if y 0 0 if 0 . Since G ( y ( 1 e 2 y if y 0 0 if 0 is the c.d.f. of an EXP ¡ 1 2 ¢ random variable, therefore Y n D Y v EXP( 1 2 ) . 1
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( c )[3 ] Show U n =1 e 2 Y n D U v UNIF (0 , 1) . Since Y n D Y v EXP( 1 2 ) then by Slutsky’s Theorem U n e 2 Y n D 1 e 2 Y = W where Y v EXP( 1 2 ) . The p.d.f. of Y is g ( y )=2 e 2 y , if y 0 . Since w e 2 y ,y = 1 2 log (1 w ) and dy dw = 1 2(1 w ) .
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This note was uploaded on 02/22/2010 for the course STAT 330 taught by Professor Paulasmith during the Fall '08 term at Waterloo.

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t309soln - STAT 330 Test 3 Solutions 1 Suppose X1 Xn is a...

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