Very often I've seen the basic density $$f(x,y)=2,\quad \text{for} \ x+y<1 \text{ where}\ x\in (0,1),\ y\in (0,1) $$
and it's relatively standard to show that $E(X)=E(Y)=\frac{1}{3}$.
I was wondering, what is the intuition for this result?
(I don't mean an intuition for the marginal of $X$ and then calculating the expectation, but a way to see possible visually why this might be the case)
The point $\langle X,Y\rangle$ is uniformly distributed over the triangle $\triangle\langle 0,0\rangle\langle 0,1\rangle\langle 1,0\rangle$.
This triangle's centroid is $\langle \tfrac 13,\tfrac 13\rangle$ - which is one third of the vector sum of the vertices.
This is the centre of gravity of a such a shape with a uniform density.
That is all.