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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2025 • Question 3

Question

Math 6010 • Linear Models Spring 2025 • Question 3

Let \((X_{1}, X_{2})\) be a bivariate normal with \(\mathbb {E}X_{1} = \mathbb {E}X_{2} = 0\) and covariance matrix \[ \begin {bmatrix} 2 & \rho \\ \rho & 2 \end {bmatrix}. \] Determine the \(\rho \) such that \(2X_{1} + X_{2}\) and \(X_{1} + 3X_{2}\) are independent.

Raw LaTeX

Let \((X_{1}, X_{2})\) be a bivariate normal with \(\mathbb{E}X_{1} = \mathbb{E}X_{2} = 0\) and covariance matrix
\[
\begin{bmatrix}
2 & \rho \\
\rho & 2
\end{bmatrix}.
\]
Determine the \(\rho\) such that \(2X_{1} + X_{2}\) and \(X_{1} + 3X_{2}\) are independent.

Topic Tags

Covariance and Independence Multivariate Normal Distributions