Hello guys, and sorry for my english in advance.


I was presented some time ago with the following problem:
Suppose there is a frog that jumps in any direction randomly, and all the jumps have size 1. What's the probability of, after 3 jumps, the frog be less than 1 unit from the origin.


I solved the problem with a double integral (if I remember well the answer is 25%), but then I thought about a similar and more general problem that I found out it's a lot similar to the Browninan motion.


Suppose there is a particle in a 2-D space that moves only in displacements of size 1 and random directions that follow a uniform probability density. What is the PDF of the particle distance from the origin after N moves? What is the probability of, after N movements, the particle end in a distance less then kN from the origin? N>>1


I approached it in the following way (and I couldn't finish):


Let θi ∈ [0, 2π)
Final position: (∑cos(θi), ∑sin(θi))
So (∑cos(θi))2 + (∑sin(θi))2 < k2N2
Define θij = θi - θj ∈ (-2π, 2π)
∑cos(θij)) < (k2 N2 - N)/2


The PDF of θij is not uniform, but if we define
αij = x if 0 ≤x < π
2π- x if π≤x < 2π


Then αij has a uniform distribution and covers all the values of cos(x), that way we can define Cij = cos(αij) with probbility density function 1/sqrt(1-x2) and


∑Cij < (k2N2-N)/2


but I do not find a way to calulate the PDF of the N (N-1)/2 variables above.


Can anyone help me?