# Probability density function of the node positions in a random walk after N time slots

Hello, my question basically is how do I find the probability density function of the position of the nodes in a given area after N discrete time slots when the nodes move following the 2D random walk model.

In case the question isn't clear, I'll just explain again. say I'm generating 50 nodes from Uniform distribution in a circular area of radius D. The initial positions(r,$\theta$) of these nodes have been generated as follows : $r$ ~Unif[0,D] and $\theta$~Unif($-\pi,\pi$]. Then each of these nodes pick up a random direction of motion in the range $\phi$~Unif($-\pi,\pi$]. I want to know the distribution of the positions of these nodes after a long time, say T time slots. Each of these time slots are of fixed length and every node moves with a fixed velocity $v$ in every time slot. Can someone help me on this? It seemed to be quite a basic problem and the solution to this should exist in the literature, may be in some other context, but I couldn't find any.

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That initial distribution isn't uniform. Also, what do you do when the nodes (is that term standard?) hit the boundary of the region? If there are no boundary effects, why can't you use a normal approximation? –  Douglas Zare Mar 11 at 14:35
Well, when I mean uniformly distributed nodes in a network area, I essentially mean that their positions have been generated from a uniform distribution. So, I said I generate radial distance $r$ from a uniform distribution between 0 to D and angular position $\theta$ from $-\pi$ to $\pi$. Didn't understand why are you calling it non-uniform. Regarding boundary effects, the nodes reflect back inside when they hit the boundary. If removing that condition helps you, assume that my nodes move on the surface of a sphere. I just want to ensure that total no. of nodes are fixed in the network area. How are you concluding that their steady state spatial distribution is going to be normal? Thanks in advance.