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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 8

Question

Math 6010 • Linear Models Spring 2009 • Question 8

Suppose \(f\) is a probability density function on \(\mathbf {R}\), and define for all \(x,y\in \mathbf {R}\), \[ g(x,y):= \begin {cases} 2f(x)f(y), & \text {if } x\leq y,\\ 0, & \text {otherwise.} \end {cases} \] (a) Prove that \(g\) is the probability density function of some random variable \((X,Y)\) which takes values in \(\mathbf {R}^2\).

(b) Are \(X\) and \(Y\) independent? Prove or disprove.

Raw LaTeX

Suppose \(f\) is a probability density function on \(\mathbf{R}\), and
define for all \(x,y\in\mathbf{R}\),
\[
  g(x,y):=
  \begin{cases}
    2f(x)f(y), & \text{if } x\leq y,\\
    0, & \text{otherwise.}
  \end{cases}
\]
\textbf{(a)} Prove that \(g\) is the probability density function of some random
variable \((X,Y)\) which takes values in \(\mathbf{R}^2\).

\textbf{(b)} Are \(X\) and \(Y\) independent? Prove or disprove.

Topic Tags

Classical Probability and Inclusion-Exclusion Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods