Dr. Hackney STA Solutions pg 74

Dr. Hackney STA Solutions pg 74 - 5-8 Solutions Manual for...

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5-8 Solutions Manual for Statistical Inference = 1 2 π ν ν/ 2 Γ( ν/ 2)2 ν/ 2 0 x (( ν +1) / 2) - 1 e - ( ν + t 2 ) x/ 2 dx integrand is kernel of gamma(( ν +1) / 2 , 2 / ( ν + t 2 ) = 1 2 π ν ν/ 2 Γ( ν/ 2)2 ν/ 2 Γ(( ν + 1) / 2) 2 ν + t 2 ( ν +1) / 2 = 1 νπ Γ(( ν +1) / 2) Γ( ν/ 2) 1 (1 + t 2 ) ( ν +1) / 2 , the pdf of a t ν distribution. b. Differentiate both sides with respect to t to obtain νf F ( νt ) = 0 yf 1 ( ty ) f ν ( y ) dy, where f F is the F pdf. Now write out the two chi-squared pdfs and collect terms to get νf F ( νt ) = t - 1 / 2 Γ(1 / 2)Γ( ν/ 2)2 ( ν +1) / 2 0 y ( ν - 1) / 2 e - (1+ t ) y/ 2 dy = t - 1 / 2 Γ(1 / 2)Γ( ν/ 2)2 ( ν +1) / 2 Γ( ν +1 2 )2 ( ν +1) / 2 (1 + t ) ( ν +1) / 2 . Now define y = νt to get f F ( y ) = Γ( ν +1 2 ) ν Γ(1 / 2)Γ( ν/ 2) ( y/ν ) - 1 / 2 (1 + y/ν ) ( ν +1) / 2 , the pdf of an F 1 . c. Again differentiate both sides with respect to t , write out the chi-squared pdfs, and collect terms to obtain ( ν/m ) f F (( ν/m ) t ) = t - m/ 2 Γ( m/ 2)Γ( ν/ 2)2 ( ν + m ) / 2 0 y ( m + ν - 2) / 2 e - (1+ t ) y/ 2 dy. Now, as before, integrate the gamma kernel, collect terms, and define y = ( ν/m ) t to get f F (
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