**tldr;** The basic essence of this note is to demonstrate the importance of vision. Instead of blindly chasing everything that is shiny, a person with vision can walk in a straight line. This is important, a person walking in a straight line with N steps moves with $O(n)$ But a person walking in a random walk, only can move on the order, $O(\sqrt{n})$.

I originally saw this assertion in, The Art of doing Science and Engineering by Richard Hamming.

Assuming there are independent random variables, $Z_1, Z_2, Z_3, ...$ such that each variable is either -1 or 1 with a 50% probability. Then create a length-N sequence such that $S_0 = 0$ and $S_N = \sum_{j=1}^{N} Z_j$.

It follows that the expected value, $E(S_N) = \sum_{j=1}^{N} E(Z_j) = 0$

Weβre going to need another property of sums to move on here,

Then, we can find:

because $E(Z_iZ_j)$ is 0, since the variables are independent and have a mean of 0. So it follows that the distance is roughly on the order of $\sqrt{N}$.