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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2007 • Question 9

Question

Math 6010 • Linear Models Fall 2007 • Question 9

Let \(X\) and \(Y\) be independent random variables with density function \(f\) and \(g\). Assume that \(Y>0\). Show that the function \[ h(t)=\int _{0}^{\infty } z f(zt)g(z)\,dz \] can be chosen as the density of \(Z=X/Y\). (7 points)

Raw LaTeX

Let \(X\) and \(Y\) be independent random variables with density
function \(f\) and \(g\). Assume that \(Y>0\). Show that the function
\[
  h(t)=\int_{0}^{\infty} z f(zt)g(z)\,dz
\]
can be chosen as the density of \(Z=X/Y\). (7 points)

Topic Tags

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