# 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 '13 at 14:35
By uniformly distributed nodes in a network area, I mean their positions are 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 you call it non-uniform. Regarding boundary effects, the nodes reflect back inside when they hit the boundary. If it helps, assume 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 do you conclude their steady state spatial distribution is normal? –  user32138 Mar 12 '13 at 14:00
In the uniform distribution on the disk, the density of small radii is lower than the density of large radii. So, if you are choosing the radius by drawing it from a uniform distribution, then you aren't choosing the point uniformly from the disk. That's why I said it wasn't uniform. I asked about the normal distribution because a reasonable question is to ask what happens in a random walk with no boundary. You didn't specify what happens on the boundary. If you do mean a region with boundary, then what happens on the boundary is important. –  Douglas Zare Mar 12 '13 at 14:36