Give such a random walk moving on the x-axis:
- Start from $x_0=0$;
- After the $i^{th}$ step, the location is $x_i$.
- The length for the $i^{th}$ step $x_i-x_{i-1}$ is a uniformly generated real number in $[-1,1]$. Negative length means to move to the positive direction.
The problem is to compute
$\text{Expectation}\left(\max\limits_{0\leq i,j \leq n}(x_i-x_j)\right)$
and
$\text{Expectation}\left(\max\limits_{0\leq i \leq j \leq n}(x_i-x_j)\right)$