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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 10

Question

Math 6010 • Linear Models Fall 2018 • Question 10

Show that for every \(\rho \in (-1,1)\) the function \[ f(x,y) = \frac {1}{2\pi \sqrt {1-\rho ^2}} \exp \left ( -\frac {x^2-2\rho xy+y^2}{2(1-\rho ^2)} \right ) \] is a pdf on \(\mathbb {R}^2\), and that if \((X,Y)\) is a pair of random variables with pdf \(f\) then \[ \mathbb {P}(XY<0)=\frac {1}{\pi }\arccos \rho . \]

Raw LaTeX

Show that for every \(\rho\in(-1,1)\) the function
\[
  f(x,y)
  =
  \frac{1}{2\pi\sqrt{1-\rho^2}}
  \exp\left(
    -\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}
  \right)
\]
is a pdf on \(\mathbb{R}^2\), and that if \((X,Y)\) is a pair of random variables
with pdf \(f\) then
\[
  \Prob(XY<0)=\frac{1}{\pi}\arccos\rho.
\]

Topic Tags

Measure-Theoretic Probability Uniform Integrability and Truncation Methods