hw2 - Homework 2 Solutions 6.50. We know that if U = (n 1)S...

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Homework 2 Solutions 6.50. We know that if U =( n 1) S 2 2 χ 2 n 1 , then E ( U )= n 1 and V ( U )=2( n 1). Using Equation (6.12) with X = U , k =1 / 2, and ν = n 1 we get E ( U )= 2Γ( n/ 2) / Γ[( n 1) / 2]. a. E ( S 2 )= E [ σ 2 U/ ( n 1)] = σ 2 E ( U ) / ( n 1) = σ 2 . b. V ( S 2 )= V [ σ 2 U/ ( n 1)] = σ 4 V ( U ) / ( n 1) 2 =2 σ 4 / ( n 1). c. E ( S )= E [ σ U/ n 1] = σE ( U ) / n 1= σ 2Γ( n/ 2) / { n 1Γ[( n 1) / 2] } 6.51. By a proposition and independence, we know M X 3 ( t )= M X 1 ( t ) M X 2 ( t ) so that M X 2 ( t )= M X 3 ( t ) M X 1 ( t ) =(1 2 t ) ν 3 / 2 (1 2 t ) ν 1 / 2 =(1 2 t ) ( ν 3 ν 1 ) / 2 which is recognized as the mgf of a χ 2 ν 3 ν 1 random variable. 6.64. a. By a proposition, X = 4 i =1 Z 2 i χ 2 4 . b. Since
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This note was uploaded on 01/23/2009 for the course STAT 319 taught by Professor Smith during the Fall '08 term at Pennsylvania State University, University Park.

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