409Hw03ans - STAT 409 Homework #3 ( due Friday, September...

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STAT 409 Homework #3 Fall 2011 ( due Friday, September 16, by 4:00 p.m. ) 1. a) Let X have a χ 2 ( r ) distribution. If k > – r / 2 , show that E ( X k ) exists and it is given by E ( X k ) = Γ + Γ 2 2 2 r k r k . E ( X k ) = ( ) ( ) - - Γ 0 2 1 2 2 2 2 1 dx e x r x x r r k = - + + - + Γ Γ + Γ 0 2 1 2 2 2 2 1 2 2 2 dx e x k r r k r x k r k r k = Γ + Γ 2 2 2 r k r k , since 2 1 2 2 2 2 1 x k r k r e x k r - + Γ - + + is the p.d.f. of χ 2 ( r + 2 k ) distribution.
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b) 6.1-14 (a), (b) ( ) a) Recall that ( n – 1 ) S 2 / σ 2 has a χ 2 ( n – 1 ) distribution. From part (a), if r = n – 1 and k = 1 / 2 , then - Γ Γ - = 2 1 2 2 S 1 E σ n n n . Therefore, σ S 2 2 2 1 1 E = Γ - Γ - n n n , and S 2 2 2 1 1 Γ - Γ - n n n is unbiased for σ . b) n = 5 c = Γ Γ 2 5 2 2 4 4 = π 2 1 2 3 2 1 2 = π 2 3 8 1.063846. n = 6 c = Γ Γ 2 6 2 2 5 5 = 2 2 2 1 2 3 5 π = 2 8 5 3 π 1.050936. 2.
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409Hw03ans - STAT 409 Homework #3 ( due Friday, September...

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