410Hw09ans - STAT 410 Homework #9 Spring 2009 (due Monday,...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STAT 410 Homework #9 Spring 2009 (due Monday, July 27, by 4:00 p.m.) 1. 4.3.16 Hint: n n e t n t n t e z n t 6 2 1 3 2 + + + = for some z between 0 and n t . a) M X 1 ( t ) = ( 1 t ) 1 , t < 1. M Y n ( t ) = ( 29 - 1 X E n n t e = ( 29 + + +- n t n t n e e 2 1 X ... X X E = n n t n t e 1 X M - = n n t n t e-- - 1 = n n t n t e n t e- - , 1 < n t . b) n n e t n t n t e u n t 6 2 1 3 2 + + + = for some u between 0 and n t . Also n e t n t e v n t 2 1 2 + + = for some v between 0 and n t . Thus, for some u between 0 and n t and for some v between 0 and n t , M Y n ( t ) = n v u n n e t n t n t n n e t n t n t- --- + + + 2 6 2 1 3 2 3 2 = n v u n n e t n n e t n t- - +- 2 6 2 1 3 3 2 . As n , M Y n ( t ) 2 2 exp t = M Z ( t ), where Z has Standard Normal N ( 0, 1 ) distribution. Z Y D n , Z ~ N ( 0, 1 ). 2. 4.3.17 Hint: Use Theorem 4.3.9. From 4.3.16, Y n = ( 29 1 X- n n D N ( 0, 1 ). Let g ( x ) = x . Then g ( x ) is differentiable, g ' ( x ) = x 2 1 , and g ' ( 1 ) = 2 1 . Then, by Theorem 4.3.9, ( 29 1 X- n n ( 29 ( 29 1 1 , 2 ' g N D = 4 1 , N . 3. 4.3.19 F ( x ) = x 5 , 0 < x < 1. F Y 1 ( x ) = 1 ( 1 x 5 ) n , 0 < x < 1. F Z n ( z ) = P ( n p Y 1 z ) = F Y 1 ( z / n p ) = n p n z 5 5 1 1 -- , 0 < z < n p . If p = 1 / 5 , F Z n ( z ) 1 e z 5 , z > 0, as n , and Z n converges in distribution. If p < 1 / 5 , as n , n p n z 5 5 1 - 0, and F Z n ( z ) 1, z > 0, and Z n P 0. If p > 1 / 5 , as n , n p n z 5 5 1 - 1, and F Z n ( z ) 0, z > 0, and Z n does not have a limiting distribution. 4. 5.1.3 Hint: Table II ( p. 673 ) gives quantiles (percentiles) of 2 distribution....
View Full Document

Page1 / 11

410Hw09ans - STAT 410 Homework #9 Spring 2009 (due Monday,...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online