I don't know what this means...Do I need the abstraction of Y1 and Y2 or can I just leave it in terms of X?

You need to make a two dimensional change.

It's just calc 3.

The joint density of X_1 and X_2 is the product of their marginals.

So, f(x_1,x_2)=exp{-(x_1+x_2)} in the irst quadrant, x_1>0, x_2>0.

NOW ge the jacobian, which you should recall is the determinant of the four partial derivative.

INSERT your new rvs, y_1 and y_2 and multiple by the absolute value of your jacobian and you're done.

Clearly, since X_1 and X_2 are exponentials, which are gamma (1,1)

We have via MGFs that the sum is a gamma (2,1)

And one can see that Y_2 is between 0 and 1, hence it's Beta with mean .5.

I think it's a Beta(1,1).

But to see that they are independent you need the joint density of Y_1 and Y_2.

, just learning your TeX here