- #1
jaumzaum
- 434
- 33
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 ##\theta_i ∈ [0, 2\pi)##
Final position: ## (∑cos(\theta_i), ∑sin(\theta_i))##
So ##(∑cos(\theta_i))^2 + (∑sin(\theta_i))^2 < k^2N^2##
Define ##\theta_{ij} = \theta_i - \theta_j ∈ (-2\pi, 2\pi)##
##∑cos(\theta_{ij})) < (k^2 N^2 - N)/2##
The PDF of ##\theta_ij## is not uniform, but if we define
##\alpha_{ij} =
\begin{cases}
x & \text{if } 0 \leq x < \pi \\
2\pi - x & \text{if } \pi \leq x < 2\pi \\
\end{cases}##
Then ##\alpha_{ij}## has a uniform distribution and covers all the values of ##cos(x)##, that way we can define ##C_{ij} = cos(\alpha_{ij})## with probbility density function ##1/sqrt(1-x^2)## and
##∑C_{ij} < (k^2N^2-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?
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 ##\theta_i ∈ [0, 2\pi)##
Final position: ## (∑cos(\theta_i), ∑sin(\theta_i))##
So ##(∑cos(\theta_i))^2 + (∑sin(\theta_i))^2 < k^2N^2##
Define ##\theta_{ij} = \theta_i - \theta_j ∈ (-2\pi, 2\pi)##
##∑cos(\theta_{ij})) < (k^2 N^2 - N)/2##
The PDF of ##\theta_ij## is not uniform, but if we define
##\alpha_{ij} =
\begin{cases}
x & \text{if } 0 \leq x < \pi \\
2\pi - x & \text{if } \pi \leq x < 2\pi \\
\end{cases}##
Then ##\alpha_{ij}## has a uniform distribution and covers all the values of ##cos(x)##, that way we can define ##C_{ij} = cos(\alpha_{ij})## with probbility density function ##1/sqrt(1-x^2)## and
##∑C_{ij} < (k^2N^2-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?