I'm trying to derive the p.m.f and p.d.f for the convolution two jointly discrete/continuous random variables $X$ and $Y$, i.e. $Z = X+Y$ and I want $P_Z(z)$ and $f_Z(z)$.
Discrete:
$$P(Z = z) = \sum_{\{(x,y)\in\mathbb{R}^2|x+y = z\}}{P(X=x,Y=y)}$$
Then using the identity:
$$\sum_{x}{P(X=x,Y=z-x)}$$ And I think that's as far as I can go without knowing they are independent.
Continuous:
I know I need to start with the c.d.f of $Z$
$$P(Z\le z)= F_Z(z) = \int_{-\infty}^{z} f_Z(u) du = \int\int_{\{(x,y)\in\mathbb{R}^2|x+y \le z\}} f_{X,Y}(x,y)dxdy = \int_{-\infty}^{\infty}\int_{\infty}^{z-y}{f_{X,Y}(x,y)dxdy}$$
Here is where I am unsure, what can I do with the limits of the double integrals. I have been told to do a substitution for $u$ and $v$ to change the limits however I am not sure how to do this. If my method is wrong please let me know :)