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Prove: Let X₁, X₂,..., X„ be a r.s. from N(µ,σ²) and Y₁,Y,₂,..., Y be another independent r.s. from N(μ₂,₂²). Then, (X-Ỹ)-(μ₁ −μ₂) t(n₁+n₂-2) assuming o = 2, 1 1 m M₂ where S = (n − 1)S² + (n₂ −1)² m₂ +1₂-2 Some hints: (n₁ - 1)S (n2 - 1)S2 (a) Determine the (marginal) distributions of and 02 02 (b) Use the fact the chi-square distribution is reproductive to find the distribution of (n₁ - 1) S² 02 (n₂ - 1)S2 + 02 ·+·

Prove: Let X₁, X₂,..., X„ be a r.s. from N(µ,σ²) and Y₁,Y,₂,..., Y be another independent r.s. from N(μ₂,₂²). Then, (X-Ỹ)-(μ₁ −μ₂) t(n₁+n₂-2) assuming o = 2, 1 1 m M₂ where S = (n − 1)S² + (n₂ −1)² m₂ +1₂-2 Some hints: (n₁ - 1)S (n2 - 1)S2 (a) Determine the (marginal) distributions of and 02 02 (b) Use the fact the chi-square distribution is reproductive to find the distribution of (n₁ - 1) S² 02 (n₂ - 1)S2 + 02 ·+·

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Prove: Let X₁, X₂,..., X be a r.s. from N(µ,σ²) and Y₁,Y₂,...,Y be another independent r.s. from n₁ N(μ₂,02³). Then, (X-Ỹ)-(μ-μ₂) t(n+n₂-2) assuming of = 2, 1 1 + P m M₂ where S = (n − 1)S² + (n₂ −1)S² n+n,-2 Some hints: (n₁ − 1) S² (a) Determine the (marginal) distributions of and (n₂ - 1)S2/ 02 02 (b) Use the fact the chi-square distribution is reproductive to find the distribution of (n₁ − 1)S² (n₂ - 1)S²/ - + 02 02
Transcribed Image Text:Prove: Let X₁, X₂,..., X be a r.s. from N(µ,σ²) and Y₁,Y₂,...,Y be another independent r.s. from n₁ N(μ₂,02³). Then, (X-Ỹ)-(μ-μ₂) t(n+n₂-2) assuming of = 2, 1 1 + P m M₂ where S = (n − 1)S² + (n₂ −1)S² n+n,-2 Some hints: (n₁ − 1) S² (a) Determine the (marginal) distributions of and (n₂ - 1)S2/ 02 02 (b) Use the fact the chi-square distribution is reproductive to find the distribution of (n₁ − 1)S² (n₂ - 1)S²/ - + 02 02
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