Probability, Random Walk, Distance from Origin after N steps in 2 and 3 dimensions I am looking for a formula of the distance from Origin after N equal steps in random directions in a 2 or 3 dimensional spaces. Can someone help me with a reference to a book, article or any publication dealing with this subject? Thanks!
note the question asks for a formula, simulating the thing is trivial..

As a first step, since you only ask for the distance, work out the incremental change in distance with each step assuming a unit step in a random direction. You can then develop a simple 1D numerical approach to find the distance after n steps: (mathematica code, the question was orignally posted to the mathematica site)

md[n_Integer] := (
       Mean@Table[
        dissquared = 0;
        Do[ 
          dissquared += 
               1 + 2 Cos[RandomReal[{0, 2 Pi}]] Sqrt[dissquared ];
         , {n}];
         Sqrt[dissquared], {2000}])


 Show[Plot[ Sqrt[Pi i/4 ], {i, 1, 500}, PlotStyle -&gt; Red], 
      ListPlot[Table[{i, md[i]}, {i, 1, 500, 10}]]]


oops I dont seem to be allowd to post a graphic.  Anyway we can emperically work out an expression for the average distance as 

d[n] = Sqrt[ n Pi /4 ]


Note obviously d[1] must be 1, but the simulation converges to the above for n larger than 4..
If someone can finish up and show that analytically I'd like to see..
2 apprioriate models for your task could be:

1) Markovian matrix with probabilities between each state.

or 2) Brownian motion