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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2008 • Question 7

Question

Math 6010 • Linear Models Fall 2008 • Question 7

Suppose \(W:=(X,Y)\) is a two-dimensional random variable and \(QW^T\) is \(N(0,1)\) for all \(2\times 2\) rotation matrices \[ Q:= \begin {pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end {pmatrix}. \] Prove that \(X\) and \(Y\) are independent and \(N(0,1)\).

Raw LaTeX

Suppose \(W:=(X,Y)\) is a two-dimensional random variable and
\(QW^T\) is \(\N(0,1)\) for all \(2\times2\) rotation matrices
\[
  Q:=
  \begin{pmatrix}
    \cos\theta & \sin\theta\\
    -\sin\theta & \cos\theta
  \end{pmatrix}.
\]
Prove that \(X\) and \(Y\) are independent and \(\N(0,1)\).

Topic Tags

Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods