two numbers chosen from 0 to 1, expected val of larger number If I pick two numbers in the interval (0,1), what is the expected value of the larger number?
Please help. I know that the expected value of one number is 0.5 but I'm not sure how to find the expected vlaue of the larger number here.

Imagine the scenario with two die:

6 6 6 6 6 6
5 5 5 5 5 6
4 4 4 4 5 6
3 3 3 4 5 6
2 2 3 4 5 6
1 2 3 4 5 6

The expected value here is:
$$\frac{1+6+15+28+45+66}{36}=\frac{161}{36}=4.47\dot2$$
$161$ is the sixth term in the sequence of hexagonal pyramid numbers ($A002412$).
These are given by:
$$\frac{n(n+1)(4n-1)}{6}$$
On the unit grid, we can do a similar trick. However if say we use a resolution of $k$, then to calculate the expectation we use $kP(X=k)$, and over $k^2$ squares, to give the result as:
$$\lim_{k\to\infty}\frac{k(k+1)(4k-1)}{6k^3}=\frac23$$
