Can someone very simply explain to me how to compute the expected distance from the origin for a random walk in $1D, 2D$, and $3D$? I've seen several sources online stating that the expected distance is just $\sqrt{N}$ where $N$ is the number of time steps, but others say that the expected distance is $\sqrt{\frac{2N}{\pi}}$. Which one is it and is it the same regardless of dimension?
Thanks
Best Answer
The expected value of the square of the absolute distance from the origin is $N$ (you are adding together $N$ independent random variables with mean $0$ and absolute magnitude $1$), and this is true in any dimension.
So those sources which are telling you $$\sqrt N$$ are giving you this as in some sense the "root mean square" distance from the starting point. It is not the expected value of the distance.
For a one dimensional random walk the expected absolute distance from the origin after $N$ steps is not easy to state explicitly, but as $N$ increases it becomes close to $$\sqrt{\dfrac{2N}{\pi}}.$$ So the sources which give you that are in a sense talking about a limit.
This changes for higher dimensions: if there are $d$ dimensions then the expected absolute distance from the origin after $N$ steps becomes close to $$\sqrt{\dfrac{2N}{d}} \dfrac{\Gamma(\frac{d+1}{2})}{\Gamma(\frac{d}{2})}$$ where $\Gamma$ is the Gamma function. As the number of dimensions increases, this get close to but is still below $\sqrt N$.
This is related to the means of the chi- and chi-squared distributions.