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Uncle's epic journey

One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently and uniformly) with an average of $1$ village per square mile. Our village is in the middle of a very large country, such that my uncle could not possibly reach the border within one year.

Here is an example of what the first two weeks of his journey could look like.

enter image description here

I haven't heard from my uncle since he left, and I don't have a map. How far away do you reckon he is now? That is,

What is the expectation of his distance (as the crow flies) from our village?

My attempt

I have only been able to get a crude approximation, $\sqrt{\frac{365}{\pi}}\approx10.78$$\frac14\sqrt{365\pi}\approx8.47$ miles, by making the following two assumptions.

  • Assume his journey is effectively just like a $2D$ random walk with constant step size $\lambda$, where $\lambda$ is defined as the expected step size of his actual journey.
  • Assume $\lambda=$ expected nearest neighbor distance in a $2D$ Poisson process with intensity $1$.

Under the second assumption, $\lambda=\frac12$. Proof: If $X$ is the distance to a village's nearest neighbor, then the density function is $f(x)=2\pi xe^{-\pi x^2}$, so $\mathbb{E}(X)=\int_0^\infty xf(x)\mathrm dx=\frac12$.

Then under the first assumption, my uncle's expected distance from our village is $2\lambda\sqrt{\frac{365}{\pi}}\approx10.78$$\frac{\lambda}{2}\sqrt{365\pi}\approx8.47$.

I suspect the first assumption causes an underestimate: my uncle never revisits a village, so his path involves less "back-tracking" than a $2D$ random walk. The second assumption definitely causes an underestimate, because usually my uncle's step size is greater than the nearest neighbor distance. So I suspect the correct answer is considerably larger than $10.78$$8.47$.

Context

Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here and here.

Uncle's epic journey

One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently and uniformly) with an average of $1$ village per square mile. Our village is in the middle of a very large country, such that my uncle could not possibly reach the border within one year.

Here is an example of what the first two weeks of his journey could look like.

enter image description here

I haven't heard from my uncle since he left, and I don't have a map. How far away do you reckon he is now? That is,

What is the expectation of his distance (as the crow flies) from our village?

My attempt

I have only been able to get a crude approximation, $\sqrt{\frac{365}{\pi}}\approx10.78$ miles, by making the following two assumptions.

  • Assume his journey is effectively just like a $2D$ random walk with constant step size $\lambda$, where $\lambda$ is defined as the expected step size of his actual journey.
  • Assume $\lambda=$ expected nearest neighbor distance in a $2D$ Poisson process with intensity $1$.

Under the second assumption, $\lambda=\frac12$. Proof: If $X$ is the distance to a village's nearest neighbor, then the density function is $f(x)=2\pi xe^{-\pi x^2}$, so $\mathbb{E}(X)=\int_0^\infty xf(x)\mathrm dx=\frac12$.

Then under the first assumption, my uncle's expected distance from our village is $2\lambda\sqrt{\frac{365}{\pi}}\approx10.78$.

I suspect the first assumption causes an underestimate: my uncle never revisits a village, so his path involves less "back-tracking" than a $2D$ random walk. The second assumption definitely causes an underestimate, because usually my uncle's step size is greater than the nearest neighbor distance. So I suspect the correct answer is considerably larger than $10.78$.

Context

Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here and here.

Uncle's epic journey

One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently and uniformly) with an average of $1$ village per square mile. Our village is in the middle of a very large country, such that my uncle could not possibly reach the border within one year.

Here is an example of what the first two weeks of his journey could look like.

enter image description here

I haven't heard from my uncle since he left, and I don't have a map. How far away do you reckon he is now? That is,

What is the expectation of his distance (as the crow flies) from our village?

My attempt

I have only been able to get a crude approximation, $\frac14\sqrt{365\pi}\approx8.47$ miles, by making the following two assumptions.

  • Assume his journey is effectively just like a $2D$ random walk with constant step size $\lambda$, where $\lambda$ is defined as the expected step size of his actual journey.
  • Assume $\lambda=$ expected nearest neighbor distance in a $2D$ Poisson process with intensity $1$.

Under the second assumption, $\lambda=\frac12$. Proof: If $X$ is the distance to a village's nearest neighbor, then the density function is $f(x)=2\pi xe^{-\pi x^2}$, so $\mathbb{E}(X)=\int_0^\infty xf(x)\mathrm dx=\frac12$.

Then under the first assumption, my uncle's expected distance from our village is $\frac{\lambda}{2}\sqrt{365\pi}\approx8.47$.

I suspect the first assumption causes an underestimate: my uncle never revisits a village, so his path involves less "back-tracking" than a $2D$ random walk. The second assumption definitely causes an underestimate, because usually my uncle's step size is greater than the nearest neighbor distance. So I suspect the correct answer is considerably larger than $8.47$.

Context

Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here and here.

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How far away is my uncle? ("Random walk" A walk on a $2D$ Poisson process, each in which every step goes to the nearest unvisited point.): expected distance from origin after $365$ steps?

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Uncle's epic journey

One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently and uniformly) with an average of $1$ village per square mile. Our countryvillage is infinitelyin the middle of a very large country, such that my uncle could not possibly reach the border within one year.

Here is an example of what the first two weeks of his journey could look like.

enter image description here

I haven't heard from my uncle since he left, and I don't have a map. How far away do you reckon he is now? That is,

What is the expectation of his distance (as the crow flies) from our village?

My attempt

I have only been able to get a crude approximation, $\sqrt{\frac{365}{\pi}}\approx10.78$ miles, by making the following two assumptions.

  • Assume his journey is effectively just like a $2D$ random walk with constant step size $\lambda$, where $\lambda$ is defined as the expected step size of his actual journey.
  • Assume $\lambda=$ expected nearest neighbor distance in a $2D$ Poisson process with intensity $1$.

Under the second assumption, $\lambda=\frac12$. Proof: If $X$ is the distance to a village's nearest neighbor, then the density function is $f(x)=2\pi xe^{-\pi x^2}$, so $\mathbb{E}(X)=\int_0^\infty xf(x)\mathrm dx=\frac12$.

Then under the first assumption, my uncle's expected distance from our village is $2\lambda\sqrt{\frac{365}{\pi}}\approx10.78$.

I suspect the first assumption causes an underestimate: my uncle never revisits a village, so his path involves less "back-tracking" than a $2D$ random walk. The second assumption definitely causes an underestimate, because usually my uncle's step size is greater than the nearest neighbor distance. So I suspect the correct answer is considerably larger than $10.78$.

Context

Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here and here.

Uncle's epic journey

One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently and uniformly) with an average of $1$ village per square mile. Our country is infinitely large.

Here is an example of what the first two weeks of his journey could look like.

enter image description here

I haven't heard from my uncle since he left, and I don't have a map. How far away do you reckon he is now? That is,

What is the expectation of his distance (as the crow flies) from our village?

My attempt

I have only been able to get a crude approximation, $\sqrt{\frac{365}{\pi}}\approx10.78$ miles, by making the following two assumptions.

  • Assume his journey is effectively just like a $2D$ random walk with constant step size $\lambda$, where $\lambda$ is defined as the expected step size of his actual journey.
  • Assume $\lambda=$ expected nearest neighbor distance in a $2D$ Poisson process with intensity $1$.

Under the second assumption, $\lambda=\frac12$. Proof: If $X$ is the distance to a village's nearest neighbor, then the density function is $f(x)=2\pi xe^{-\pi x^2}$, so $\mathbb{E}(X)=\int_0^\infty xf(x)\mathrm dx=\frac12$.

Then under the first assumption, my uncle's expected distance from our village is $2\lambda\sqrt{\frac{365}{\pi}}\approx10.78$.

I suspect the first assumption causes an underestimate: my uncle never revisits a village, so his path involves less "back-tracking" than a $2D$ random walk. The second assumption definitely causes an underestimate, because usually my uncle's step size is greater than the nearest neighbor distance. So I suspect the correct answer is considerably larger than $10.78$.

Context

Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here and here.

Uncle's epic journey

One year ago, my uncle set off from our village on an epic journey, in which every day he travels to the nearest unvisited village and stays there for the night. The villages in our country are randomly located (independently and uniformly) with an average of $1$ village per square mile. Our village is in the middle of a very large country, such that my uncle could not possibly reach the border within one year.

Here is an example of what the first two weeks of his journey could look like.

enter image description here

I haven't heard from my uncle since he left, and I don't have a map. How far away do you reckon he is now? That is,

What is the expectation of his distance (as the crow flies) from our village?

My attempt

I have only been able to get a crude approximation, $\sqrt{\frac{365}{\pi}}\approx10.78$ miles, by making the following two assumptions.

  • Assume his journey is effectively just like a $2D$ random walk with constant step size $\lambda$, where $\lambda$ is defined as the expected step size of his actual journey.
  • Assume $\lambda=$ expected nearest neighbor distance in a $2D$ Poisson process with intensity $1$.

Under the second assumption, $\lambda=\frac12$. Proof: If $X$ is the distance to a village's nearest neighbor, then the density function is $f(x)=2\pi xe^{-\pi x^2}$, so $\mathbb{E}(X)=\int_0^\infty xf(x)\mathrm dx=\frac12$.

Then under the first assumption, my uncle's expected distance from our village is $2\lambda\sqrt{\frac{365}{\pi}}\approx10.78$.

I suspect the first assumption causes an underestimate: my uncle never revisits a village, so his path involves less "back-tracking" than a $2D$ random walk. The second assumption definitely causes an underestimate, because usually my uncle's step size is greater than the nearest neighbor distance. So I suspect the correct answer is considerably larger than $10.78$.

Context

Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here and here.

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Dan
Dan
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Dan
Dan
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  • 145