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Hw09updated - Does the sequence X n converge in probability...

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STAT 410 Homework #9 Spring 2008 (due Monday, March 31, by 4:30 p.m.) No credit will be given without supporting work. Warm-up: 4.2.3 1. 4.3.2 2. 4.3.3 3. 4.3.4 Hint: F Y 2 ( x ) = ( ) ( ) ( ) ( ) ° = - - ± ± ² ³ ´ ´ µ n i i n i x x i n 2 F 1 F = ( ) ( ) ( ) ( ) ( ) ( ) 1 F 1 F F 1 1 - - - - - n n x x n x . 4. a) 4.3.9 Hint: We already know that ( ) ( ) 1 , 0 2 Y N D n n n - . b) Find P ( 40 < X < 60 ) , where X has a ( ) 50 2 ° distribution. Hint: Use integration by parts 24 times or EXCEL: = CHIINV ( α , v ) gives ( ) v 2 ± ° for 2 ° distribution with v degrees of freedom, x s.t. P ( ( ) v 2 ° > x ) = α . = CHIDIST ( x , v ) gives the upper tail probability for 2 ° distribution with v degrees of freedom, P ( ( ) v 2 ° > x ) . 7. 4.3.18 8. 4.3.19
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9. a) Suppose P ( X n = 0 ) = n 1 1 - and P ( X n = n ) = n 1 , n = 1, 2, 3, … . Does the sequence { X n } converge in distribution to some random variable?
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Unformatted text preview: Does the sequence { X n } converge in probability to some random variable? b) Prove or give a counterexample: If X X P n → , then E ( X n ) → E ( X ). Cool-down: 4.3.7 Hint: ( ) ( ) α β 1 1 M Gamma t t-= , t < 1 / . _________________________________________________________________________ If you are registered for 4 credit hours: ( to be handed in separately ) (due Wednesday, April 2, by 3:00 p.m.) 10. Let Y n be χ 2 ( n ). What is the limiting distribution of Z n = n n Y-? Hint: We already know that Y n / n P → 1 and that ( ) ( ) 1 , 2 Y N D n n n →-....
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