Suppose X and Y are continuous random variables with joint PDF ((x, y) and suppose U and V are random variables that are functions of X and Y such that the transformation
X = x(U, V) and Y = y(U, V)
is one-to-one. Show that the joint PDF of U and V is
g(u, v) = ( (x(u, v), y(u, v)) ( J (u, v) (
Let R be a region in the xy-plane and let S he its preimage. Show that P((X, Y) ( R) = P((U, V) ( S) and get a double integral for each of these.