I was just introduced to the concept of a random walk while reading the Feynman lectures on physics, Volume 1. There was something in the explanation there that confused me, so I tried looking online for other explanations, and encountered the same difficulty.
I'm trying to understand how on earth the expected distance traveled could ever be the square root of the number of steps.
Here's my thinking process. I'm curious where I'm going wrong.
Suppose you start at 0, and at each step, either go forward (+1), or backward (-1), with equal probability. Suppose that you do 100 steps. Let us call X the number where you end up.
I can see how if you were to repeat this experiment many many times, the mean value of X would be 0.
But this is not interesting. What is interesting is the total distance moved, not the final displacement.
Well, it's as obvious as pie that that total distance moved is simply 100!
This is because the distance moved with each step is the absolute value of the incremental displacement at each step, which, in our example, is 1. So the total distance moved is 1 multipled by the number of steps, which is 100.
Now all the guides I've read say that using the absolute value is computationally tricky/impossible/inefficient, so it's better to first square the individual displacements, and then square root them at the end. But the problem with this is that you end up with something completely different. In our example, you'd end up with a value of 10, which is clearly incorrect if what you are interested in is the total distance traveled.
So, if we define the root mean squared distance as "the square root of the expected total distance traveled", that is fine and consistent. But to say that the root mean squared value is the expected total distance traveled seems nonsensical to me.
Yet we have statements that appear to say just that.
Since sqrt(d2) is something like the average positive distance away from 0 after N steps (technically, it's called the "root-mean-squared" distance), we expect that after N steps, the black dot will be roughly sqrt(N) steps away from where it started. So for 25 steps, we expect the black dot to have moved roughly 5 total spaces from 0 in either direction. Of course sometimes it will move more and sometimes fewer total spaces, but 5 is roughly what we might expect.
The second thing that confuses me is that this is portrayed as some sort of probabilistic function. According to my thinking descibed above (which may be woefully misguided!) both the total distance and the square root of the total distance will be exactly N and square root of N, each and every time, in the case where the magnitude of each incremental displacement is always the same (i.e. "1" in this example). So why even frame this as an expected value. How on earth could the total distance traveled be anything but N x |stepsize|?