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A simulation of 50,000 iterations gives the average distance after a 2-step (unit step) random walk on a 2 dimensional plane, which is around 1.27. But how can one mathematically prove this?

Any insight is highly appreciated!

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  • $\begingroup$ Are you just walking on the usual lattice? Makes a significant difference in the result. $\endgroup$ – lulu Apr 6 at 23:16
  • $\begingroup$ Are your steps always north, east, south, or west, or can they be in any direction? My answer assumes any direction, while kimchi lover assumed the other. $\endgroup$ – Ross Millikan Apr 6 at 23:16
  • $\begingroup$ The question assumes walking in any direction. I'm sorry for the confusion and thank you to anyone who contributed ideas/thoughts/comments! $\endgroup$ – CminorDmax Apr 7 at 1:59
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You can rotate the coordinates so the first step is to $(1,0)$. If the second step is at angle $\theta$, the end point is then $(1+\cos \theta,\sin \theta)$. The distance from the origin is then $\sqrt{(1+\cos \theta)^2+\sin^2\theta}=\sqrt{2+2\cos \theta}$ The average of this over $\theta$ is $$\frac 1{2\pi}\int_0^{2\pi}\sqrt{2+2\cos \theta}d\theta=\frac 8{2\pi}\approx 1.2732$$

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  • $\begingroup$ I think this is more likely to be what the OP had in mind than my the model I used. $\endgroup$ – kimchi lover Apr 6 at 23:20
  • $\begingroup$ That's so smart! Thank you! $\endgroup$ – CminorDmax Apr 7 at 2:04
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I get $\frac{0+2\sqrt 2 + 2} 4 \approx 1.207$ as the expected distance. The first step moves you to distance $1$ from the origin. The next step goes back to the origin with probability $1/4$, goes sideways with probability $1/2$, and goes in the same direction with probability $1/4$.

I suppose the discrepancy between what I calculate and what you report is due to Monte Carlo sampling error.

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Assuming you are describing a walk on a two dimensional grid.

First step in any direction. Second step has four choices (assume equally probable), left, right, forward, and backward. The four eqiproable distances from start are $ \sqrt{2}, \sqrt{2},2,0$. The average is $1..2071$

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