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imagineerThat
imagineerThat
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It appears that in a 1D random walk the expected RMS distance from the origin is some positive value with any positive number of steps (N):

sqrt(N)/2

However, in this same walk, the expected deviation of the fraction of steps to any one direction tends towards zero.

1/(2 * sqrt(N))

I cannot intuitively see why the former doesn't tend towards zero as well if the average steps to the left cancels with the steps to the right.

Does anyone have an intuitive explanation for this?

It appears that in a 1D random walk the expected RMS distance from the origin is some positive value with any positive number of steps:

sqrt(N)/2

However, in this same walk, the expected deviation of the fraction of steps to any one direction tends towards zero.

1/(2 * sqrt(N))

I cannot intuitively see why the former doesn't tend towards zero as well if the average steps to the left cancels with the steps to the right.

Does anyone have an intuitive explanation for this?

It appears that in a 1D random walk the expected RMS distance from the origin is some positive value with any positive number of steps (N):

sqrt(N)/2

However, in this same walk, the expected deviation of the fraction of steps to any one direction tends towards zero.

1/(2 * sqrt(N))

I cannot intuitively see why the former doesn't tend towards zero as well if the average steps to the left cancels with the steps to the right.

Does anyone have an intuitive explanation for this?

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imagineerThat
imagineerThat
  • 150
  • 9

Random walk: Need intuition behind the expected distance from origin versus expected deviation of the fraction of left/right steps.

It appears that in a 1D random walk the expected RMS distance from the origin is some positive value with any positive number of steps:

sqrt(N)/2

However, in this same walk, the expected deviation of the fraction of steps to any one direction tends towards zero.

1/(2 * sqrt(N))

I cannot intuitively see why the former doesn't tend towards zero as well if the average steps to the left cancels with the steps to the right.

Does anyone have an intuitive explanation for this?