Exercise 6.55 Let X and Y be random variables with joint density function f (x, y) =

(

1 4 e−1 2 (x+y) if x, y > 0, 0 otherwise.

Show that the joint density function of U = 1 2 (X − Y ) and V = Y is fU,V (u, v) =

(

1 2 e−u−v if (u, v) ∈ A, 0 otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fU(u) = 1 2 e−|u| for u ∈ R.