How to find sum of random vectors (continuous and discrete)?

Question

After a search on the web I found this can be done with convolutions, though there was no such a topic in our class. Is there other way to solve this?

Answer 1

Any solution is going to be using convolutions, you just don't need to call them that.

For example, in problem 4, $Z$ could be equal to $-1, 1, 2, 3,$ or $4$. For each one of those, you compute the probability $P(Z=z)$ by summing over all the possible cases that would lead to that, i.e. $(X,Y)=(x,y),\; x+y = z$. In this case the only nontrivial one is $$P(Z=1) = P((X,Y) = (0,1)) + P((X,Y) = (2,-1)).$$

Problem 5 is the same principle, but continuous. The CDF of $Z$, $G(z)$, is given by the integral of the given distribution over the region $x+y \le z$, which is to say the area of that region intersected with the unit square. The PDF of $Z$ is the derivative of that, which will come out to be the length of the line $x+y=z$ intersected with the square.