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PDF of random motion  Similar to Browninan motion (a frog jumps randomly...)
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 2D 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 θ_{i} ∈ [0, 2π)
Final position: (∑cos(θ_{i}), ∑sin(θ_{i}))
So (∑cos(θ_{i}))^{2} + (∑sin(θ_{i}))^{2} < k^{2}N^{2}
Define θ_{ij} = θ_{i}  θ_{j} ∈ (2π, 2π)
∑cos(θ_{ij})) < (k^{2} N^{2}  N)/2
The PDF of θ_{ij} is not uniform, but if we define
α_{ij} = x if 0 ≤x < π
2π x if π≤x < 2π
Then α_{ij} has a uniform distribution and covers all the values of cos(x), that way we can define C_{ij} = cos(α_{ij}) with probbility density function 1/sqrt(1x^{2}) and
∑C_{ij} < (k^{2}N^{2}N)/2
but I do not find a way to calulate the PDF of the N (N1)/2 variables above.
Can anyone help me?
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