Math 6010 • Linear Models

Let \(X\) and \(Y\) be independent standard normals and consider
\((U,V)=(aX+bY,cX+dY)\). Assume that at least one of \(a\) and \(b\) is nonzero, and
at least one of \(c\) and \(d\) is nonzero. (Otherwise either \(U\) or \(V\) would be
0 and the problem is not interesting.)

\textbf{5a)} (5 points) Show that if \(ad=bc\), then \(U\) and \(V\) cannot be
independent random variables.

\textbf{5b)} (5 points) Assume \(ad\ne bc\) and find the joint pdf of \((U,V)\).

\textbf{5c)} (5 points) Find a necessary and sufficient condition for \(U\) and
\(V\) to be independent.