(Steps Shown) Consider the data matrix X in Exercise 3.1. We have n=3 observations on p=2 variables X_1 and X_2. Form the linear
Question: Consider the data matrix \(\mathbf{X}\) in Exercise 3.1. We have \(n=3\) observations on \(p=2\) variables \(X_{1}\) and \(X_{2}\). Form the linear combinations
\[\begin{aligned} &\mathbf{c}^{\prime} \mathbf{X}=\left[\begin{array}{ll} -1 & 2 \end{array}\right]\left[\begin{array}{l} X_{1} \\ X_{2} \end{array}\right]=-X_{1}+2 X_{2} \\ &\mathbf{b}^{\prime} \mathbf{X}=\left[\begin{array}{ll} 2 & 3 \end{array}\right]\left[\begin{array}{l} X_{1} \\ X_{2} \end{array}\right]=2 X_{1}+3 X_{2} \end{aligned}\]
- Evaluate the sample means, variances, and covariance of \(\mathbf{b}^{\prime} \mathbf{X}\) and \(\mathbf{c}^{\prime} \mathbf{X}\) from first principles. That is, calculate the observed values of \(\mathbf{b}^{\prime} \mathbf{X}\) and \(\mathbf{c}^{\prime} \mathbf{X}\), and then use the sample mean, variance, and covariance formulas.
- Calculate the sample means, variances, and covariance of \(\mathbf{b}^{\prime} \mathbf{X}\) and \(\mathbf{c}^{\prime} \mathbf{X}\) using \((3-36)\). Compare the results in (a) and (b).
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