Let X1, . . . , Xn ∼ N (μ, σ2) be i.i.d. where μ is known, but σ2 is unknown. Consider the estimate σ2 = 1/n sum of (xi-u)^2 . Show that the sum of ((xi-u)/o)^2 has a chi squared distribution with n degrees of freedom. For α ∈ (0,1), denote the α-quantile of χ2(n) by qn(α). Using this notation, find numbers an,bn ∈ R such that P(σ ̃2 < anσ2) = 2.5% and P(σ ̃2 > bnσ2) = 2.5%.

Let X1, . . . , Xn ∼ N (μ, σ2) be i.i.d. where μ is known, but σ2 is unknown. Consider the estimate σ2 = 1/n sum of (xi-u)^2 . Show that the sum of ((xi-u)/o)^2 has a chi squared distribution with n degrees of freedom. For α ∈ (0,1), denote the α-quantile of χ2(n) by qn(α). Using this notation, find numbers an,bn ∈ R such that P(σ ̃2 < anσ2) = 2.5% and P(σ ̃2 > bnσ2) = 2.5%.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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