Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2019 • Question 8

Question

Math 6010 • Linear Models Fall 2019 • Question 8

Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables, and let \(Z=X+Y\).

  1. Show that \(\mathbb {E}[Z\mid X>0,Y>0]=2\sqrt {2/\pi }\).
  2. Find the distribution and the density of \(Z\) given that \(X>0\) and \(Y>0\). (You can use the CDF of a standard normal, \(\Phi (x)=\mathbb {P}(X\leq x)\), in your expression.)

Raw LaTeX

Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables, and let
\(Z=X+Y\).
\begin{enumerate}
\item[(a)] Show that \(\E[Z\mid X>0,Y>0]=2\sqrt{2/\pi}\).
\item[(b)] Find the distribution and the density of \(Z\) given that \(X>0\) and
\(Y>0\). (You can use the CDF of a standard normal,
\(\Phi(x)=\Prob(X\leq x)\), in your expression.)
\end{enumerate}

\newpage

Topic Tags

Central Limit Theorems Covariance and Independence Multivariate Normal Distributions Uniform Integrability and Truncation Methods