Section 3d

3.4 Distribution Theory

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Problem 3d.1

Given Y1,Y2,…,YnY_{1}, Y_{2}, \ldots , Y_{n} independently distributed as N(θ,σ2)N\left(\theta , \sigma^{2}\right), use Theorem 3.5 to prove that:

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(a)

Yˉ\bar{Y} is statistically independent of Q=∑i(Yi−Yˉ)2Q = \sum_{i}\left(Y_{i}-\bar{Y}\right)^{2}.

(b)

Q/σ2∼χn−12Q / \sigma^{2} \sim \chi_{n-1}^{2}.

Problem 3d.2

Use Theorem 2.5 to prove that for the full-rank regression model, RSS is independent of (β^−β)TXTX(β^−β)(\widehat{\mathbf{\beta }}-\mathbf{\beta })^{T} \mathbf{X}^{T} \mathbf{X}(\widehat{\mathbf{\beta }}-\mathbf{\beta }).

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