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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 6

Question

Math 6010 • Linear Models Fall 2017 • Question 6

Let \(X_1,\ldots ,X_m\) be \(m\geq 2\) i.i.d. standard normal random variables, and define the random vector \[ \mathbf {X}:= \begin {bmatrix} X_1\\ \vdots \\ X_m \end {bmatrix}. \] Prove that the distribution of \(\mathbf {X}\) is the same as the distribution of \(A\mathbf {X}\) for every \(m\times m\) orthogonal matrix \(A\) that is non random.

Raw LaTeX

Let \(X_1,\ldots,X_m\) be \(m\geq2\) i.i.d. standard normal random
variables, and define the random vector
\[
  \mathbf{X}:=
  \begin{bmatrix}
    X_1\\
    \vdots\\
    X_m
  \end{bmatrix}.
\]
Prove that the distribution of \(\mathbf{X}\) is the same as the distribution of
\(A\mathbf{X}\) for every \(m\times m\) orthogonal matrix \(A\) that is non random.

Topic Tags

Central Limit Theorems Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods