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 But a person walking in a random walk, only can move on the order, .
I originally saw this assertion in, The Art of doing Science and Engineering by Richard Hamming.
Assuming there are independent random variables, such that each variable is either -1 or 1 with a 50% probability. Then create a length-N sequence such that and .
It follows that the expected value,
We’re going to need another property of sums to move on here,
Then, we can find:
because 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 .