Functionn of two random variables.

Could anyone help me with this question please?

Given X and Y are continuous random variables with the following joint probability density function:

$\displaystyle

f(x,y) =

\begin{cases}

{\theta}^2 e^{-{\theta} y} & 0 < x < y \\

0 & \text{otherwise}

\end{cases}

$

i) Obtain the joint probability density function of X and Y − X.

ii) Show that X and Y − X are independent random variables and write down their (marginal)

probability density functions. Identify these distributions

For part (i)

I let U = X and Y = V - X,

My Jacobian value was 1,

and then I got $\displaystyle f(u,v) = {\theta}^2 e^{-{\theta} , (v-u)} & \text ,u > 0, (v-u) > u $ Is this the correct joint p.d.f for X and Y-X?

Also for the marginal of X, I got $\displaystyle f X(x) = e^{-{\theta x} $ but I'm not quite sure.