Section 1a

1.1 Notation, 1.2 Statistical Models, 1.3 Linear Regression Models, 1.4 Expectation and Covariance Operators

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Problem 1a.1

Prove that if a is a vector of constants with the same dimension as the random vector X\mathbf{X}, then

E[(X−a)(X−a)T]=Var⁡[X]+(E[X]−a)(E[X]−a)T \mathbb {E}\left[(\mathbf{X}-\mathbf{a})(\mathbf{X}-\mathbf{a})^{T}\right] = \operatorname {Var}\left[\mathbf{X}\right]+(\mathbb {E}\left[\mathbf{X}\right]-\mathbf{a})(\mathbb {E}\left[\mathbf{X}\right]-\mathbf{a})^{T}

If Var⁡[X]=Σ=(σij)\operatorname {Var}\left[\mathbf{X}\right] = \mathbf{\Sigma } = \left(\sigma_{i j}\right), deduce that

E[∥X−a∥2]=∑iσii+∥E[X]−a∥2 \mathbb {E}\left[\left\| \mathbf{X}-\mathbf{a}\right\| ^{2}\right] = \sum _{i} \sigma _{i i}+\left\| \mathbb {E}\left[\mathbf{X}\right]-\mathbf{a}\right\| ^{2}
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Problem 1a.2

If X\mathbf{X} and Y\mathbf{Y} are m×1m \times 1 and n×1n \times 1 vectors of random variables, and a\mathbf{a} and b are m×1m \times 1 and n×1n \times 1 vectors of constants, prove that

Cov⁡[X−a,Y−b]=Cov⁡[X,Y] \operatorname {Cov}\left[ \mathbf{X}-\mathbf{a}, \mathbf{Y}-\mathbf{b} \right] = \operatorname {Cov}\left[ \mathbf{X}, \mathbf{Y} \right]
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Problem 1a.3

Let X=(X1,X2,…,Xn)T\mathbf{X} = \left(X_{1}, X_{2}, \ldots , X_{n}\right)^{T} be a vector of random variables, and let Y1=X1,Yi=Xi−Xi−1(i=2,3,…,n)Y_{1} = X_{1}, Y_{i} = X_{i}-X_{i-1}(i = 2,3, \ldots , n). If the YiY_{i} are mutually independent random variables, each with unit variance, find Var⁡[X]\operatorname {Var}\left[\mathbf{X}\right].

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Problem 1a.4

If X1,X2,…,XnX_{1}, X_{2}, \ldots , X_{n} are random variables satisfying Xi+1=ρXi(i=1,2,…,n−1X_{i+1} = \rho X_{i} \quad (i = 1,2, \ldots , n-1), where ρ\rho is a constant, and Var⁡[X1]=σ2\operatorname {Var}\left[X_{1}\right] = \sigma^{2}, find Var⁡[X]\operatorname {Var}\left[\mathbf{X}\right].

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