Mean squared distance traveled by an unbiased random walker in 1-D?

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Discussion Overview

The discussion revolves around the mean squared distance traveled by an unbiased random walker in one dimension. Participants explore the mathematical relationships involved, particularly focusing on the second moment and variance in the context of random walks.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant presents equations for the expected position and standard deviation of a random walker, questioning the expression for the mean squared distance .
  • Another participant seeks clarification on the variable "l" in the expression "l sqrt(N)," identifying it as the step size.
  • A later reply notes that the second moment is proportional to l^2 and mentions its role in calculating variance.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the exact expression for in the context of an unbiased random walk, and multiple interpretations of the relationships involved remain evident.

Contextual Notes

The discussion includes assumptions about the definitions of variables and the relationships between moments and variance, which are not fully resolved.

CrimsonFlash
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Hey!

I've been doing some research on random walks. From what I have gathered, a random walker in 1-D will have:
<x> = N l (2 p - 1)

σ = 2 l sqrt[N p (1 - p) ]

Here, N is the number of steps, p is the probability to take a step to the right and l is the step size.
I was wondering what <x^2> would be. From what I found, it seems to be l sqrt(N) but when I try to use <x^2> = σ^2 + <x>^2 , I don't get l sqrt(N) . I would like to know what <x^2> really is for an unbiased random walk.

Thanks
 
Last edited:
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In "l sqrt(N)", what is "l"?
 
mathman said:
In "l sqrt(N)", what is "l"?

l is the step size here.
 
The second moment is proportional to l^2. In general its main use is as an intermediate to get the variance.
 

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