Distributions - sum of random variables

Question

A point $(X,Y)$ is chosen uniformly at random from the unit square ($(X,Y)\in [0,1]^2$) and the random variables are $X$ - the x coordinate and $Y$ - the y coordinate. Now of course X and Y have uniform distributions but how can we find the density of the random variable $X+Y$? I've been struggling on this.

Answer 1

There are various methods to derive this.

The graphical method is simply to note that the density function $f_{X+Y}(z)$ will be proportional to the length of the intersection of the line $x+y=z$ and the interior of the unit square.

$$f_{X+Y}(z) = z\,\mathbf 1_{0\leq z< 1}+(2-z)\,\mathbf 1_{1<z\leq 2}$$