Joint distribution ... Help!

[lenhhoxung]

I have 1 previous exam sample question:

(X,Y) has the following joint probability density function

f(x,y)= 3(x+y) when x>0, y>0, and x+y<1
0 otherwise

1. Find the probability distribution function F(x,y)
2. Find the probability density function of U=X+Y and V=2X+Y

I find troubles to solve this question, especially part 2 where the domains of U and Y are very confusing to me.

Could someone please help?

[BGM]

So you first solve the linear system and express in terms of where the linear system is given as

One usual approach is that you substitute the variables in the original inequalities defining the support of : . Then you will get a set of inequalities of to define the support/domain of .

Another nice thing is that the transformation is linear - so from the given matrix there is some geometrical interpretation from it.