Many times I see questions in the following format:
Let $X$ and $Y$ be two independent random variables with the following joint distribution
$$f_{X,Y}(x,y) = \begin{cases} d, & a\le x, y \le b \\ 0, & \text{otherwise} \end{cases}$$
Compute the probability density function of $X + Y$.
or compute $X + Y \le 4$, or $XY$, or $X + Y^2$, or $X - Y$, etc.
My question is in all of these cases, is the thing I am being asked to compute, for example $X + Y$, itself a joint distribution? I know this sounds very simplistic but I am genuinely confused, as I thought we already had the joint distribution, in my example, $d$? Is this some sort of new joint distribution that we are making out of the old joint distribution, or is it called something else?
For exmaple, $f_{XY}(x,y)$ is a 2-dim joint density that can give you a 1-dim (marginal) density $f_X(x)$ for $X$ (as if $Y$ doesn't exist), via $f_X(x) = \int f_{XY}(x,y) dy$. Similarly, it can give you a 1-dim density $f_Y(y)$ for $Y$ (as if $X$ doesn't exist).
Now consider another $W \equiv X+Y$. It is a single random variable which should have its own 1-dim density $f_W(w)$. You can obtain this via performing variable transformation (as in calculus) on $f_{XY}(x,y) \mapsto f_{WZ}(w,z)$, where $Z$ is another random variable that is auxiliary. From this new joint density you will get the desired $f_W(w) = \int f_{WZ}(w,z) dz$ as if $Z$ never existed in the first place.
Here your $W = X+Y$ happens to be like change of coordinates, and the accompanying $Z$ is often conveniently chosen to be $Z \equiv X- Y$. Of course, the choice of $Z$ is arbitrary, and $Z = X$ or $Z = Y$ are both good.
There are other ways to obtain $f_W(w)$ besides integrating out the auxiliary variable from a 2-dim joint density. The point is, you can visualize in this case that $W$ is a new coordinate axis (just like $X$) that points along the diagonal direction. Just like how $X$ has its own 1-dim density $f_X$, this $W$ has its own density $f_W$.