I'm trying to understand how transformation techniques works in case of distribution with 2 RVs. As far as i am concerned there are only few options to use one of the transformation technique:
Convolution method. We use it if we have some new RV, such as Z = X+Y then we can apply this formula: $${f_{Z}(z)=\int \limits _{-\infty }^{\infty }f_{XY}(x,z-x)~dx}$$
Jacobian method. We use it mostly If we need to change 2 RVs. So we can apply this formula below $$ f_{Z1,Z2}(z1,z2) = f(X(z1,z2), Y(z1,z2))*|J|$$ where X,Y are known RV, Z1, Z2 are new RVs and invertible and J is Jacobian.
Questions:
- Can i reuse somehow Convolution method when i need to find arbitrary RV Z=g(x,y)? I saw an example with Z = Y/X, but i don't understand how it works
- Why in some cases Jacobian method works then we transform only 1 RV, but in the others it's not?
Task: Suppose we have iid RV X,Y with PDF $$ e^{-x}, (x>=0) $$ and we need to find new PDF's for transformed RV's:
a) Z = X+Y, b) Z = X/Y c) Z = X-Y
I tried to solve this task, but none of my answers are correct
I would really appreciate it if someone could help me sort out this topic, as I am really overworked right now.