# 501 - 5.3.2. The Student’s t and Snedecor’s F...

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Unformatted text preview: 5.3.2. The Student’s t and Snedecor’s F Distributions . If X 1 , ..., X n are i.i.d. ∼ N ( μ,σ 2 ), then μ is estimated by X , with error X- μ , and standardized error T = X- μ S/ √ n . Question: Distribution of T ? Notice that: T = X- μ S/ √ n ∼ t n- 1 = X- μ σ/ √ n radicalBig ( n- 1) S 2 σ 2 ( n- 1) . Def. Let T = Z/ radicalbig Y/p , where Z ⊥ Y , Z ∼ N (0 , 1) and Y ∼ χ 2 ( p ). We say T ∼ t distribution with degree of freedom p , denoted by T ∼ t p . Its density function is f T ( t ) = Γ( p +1 2 ) Γ( p 2 ) 1 ( pπ ) 1 / 2 1 (1 + t 2 /p ) ( p +1) / 2 . Remark. f T is derived from f Z,Y ( x,y ) = f Z ( x ) f Y ( y ) via ( T,V ) = g ( Z,Y ), where T = Z/ radicalbig Y/p and V = Y ( | J | = radicalbig y/p ). Q: E ( T ), V ( T ) = ? E ( T ) = E ( Z ) E (1 / radicalbig Y/p ) = 0. V ( T ) = V ( Z ) V (1 / radicalbig Y/p ) ? Homework. 5.18 If X 1 , ..., X n are i.i.d. ∼ N ( μ 1 ,σ 2 1 ) ( X = ( X 1 , ..., X n )), Y 1 , ..., Y m are i.i.d. ∼ N ( μ 2 ,σ 2 2 ), and X ⊥...
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## This document was uploaded on 12/03/2011.

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501 - 5.3.2. The Student’s t and Snedecor’s F...

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