Math 422 Midterm 2 Review

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1. Suppose that \(X_1 \sim \on{Norm}(0,2)\) and \(X_2 \sim \on{Norm}(0,3)\). If \(\on{Cov}(X_1,X_2) = -4\), then what is the covariance matrix for the random vector \(X = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix}\)?

2. What is the joint density function for the random vector \(X\) above?

3. Which of the following possible values for the random vector \(X\) above are is a more likely outcome? \(X = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\) or \(X = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\)?

4. Suppose that \(Y = \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ 0 & 1 \end{bmatrix} X\) where \(X\) is the random vector above. What is the covariance matrix for \(Y\)?

5. The matrix equation \[\begin{bmatrix} 1 & -2 \\ 1 & -1 \\ 1 & 1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} b_0 \\ b_1 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \\ 0 \\ 1 \end{bmatrix}\] is impossible to solve. Find the least squares solution.

6. The least squares solution from the last problem corresponds to a regression line for a collection of points in \(\R^2\). Draw a graph showing the regression line and the points it approximates.

7. What are the residuals for the regression line above?

8. Suppose that \(X b = y\) is a linear equation with no solutions. Assume that the columns of \(X\) are linearly independent.

1. What matrix would you multiply \(y\) by in order to find the least squares solution \(b\).
2. What matrix would you multiply \(y\) by in order to find the vector \(\hat{y}\) in the column space of \(X\) that is closest to \(y\)?

9. For \(X = \begin{bmatrix} 1 & -2 \\ 1 & -1 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}\), what are the domain and codomain of the linear transformation defined by \(X\)?

10. What are the dimensions of the four fundamental subspaces of \(X\)?