# Feynman - Random Walk <D> and coin flipping

1. May 30, 2010

### QED-Kasper

Hello,

I have read the probability chapter in Feynman's lectures on physics. And got fascinated by the random walk. There is a statement, that in a game where either a vertical distance of +1 or -1 can be walked each move, the expected value of the absolute distance (lets call it <D>) from initial position 0, will be equal to the square root of N if N moves have been made.

For those that don't know and are interested: http://en.wikipedia.org/wiki/Random_walk.

What was fascinating for me for some reason was the fact that this expected distance <D> was becoming ever greater the more moves were made. For some reason I was thinking that the more moves the more likely the person will be at 0.

While I was thinking of this, the ordinary coin-flipping game came to my head. And I perceived an analogy. The more coins you flip the more likely that the fractional amount of tails you get will be closer to 1/2. Which is the probability of getting tails. However as the fractional amount of tails you get comes closer to 1/2, the difference between the amount of coins and tails on the average becomes bigger. Like this: 10 coin flips 4/10 tails 6/10 heads. the difference is only 2. but the fractional amount of tails is 4/10. Compared to 496 333/1000000 tails and 503777/1000000 heads. The fractional amount of tails is much closer to 1/2 but the difference between the amount of tails and heads is several thousands. So on the average you will see much greater difference between the amount of coins and tails the more you throw.
This is my question:
Isn't the average difference the same as the expected value <D> of the random walk?

Thanks for allowing me to share my experience.

2. May 31, 2010

### tiny-tim

Hello QED-Kasper!
That's right!

The coin-difference after n flips and the walk-distance after n steps (in 1D) have the same distribution … each process is a model for the other, and in particular, they have the same expected values.

3. May 31, 2010

### QED-Kasper

Thanks, I appreciate that. And thank you for being extra kind :).

4. Jun 1, 2010

### codelieb

I would like to point out two things here:

1. "Walk-distance" D (generally refered to as "distance from the origin" in the theory of random walks) is defined to be the difference between the number of 'heads' and the number of 'tails' in a (Bernoulli) sequence of 'coin flips,' while the terms "expected value" and "average" have precisely the same meaning. So, "the average difference between heads and tails" and "the expected value of D" are just two ways of saying exactly the same thing. (Stating that "each process is a model for the other" having "the same distribution" and "the same expected values" obscures the fact that they are one and the same process.)
2. The lecture on Probability in The Feynman Lectures on Physics Volume I, as well as the lecture that precedes it on Time and Distance, were written and delivered by Matthew Sands - Feynman had nothing to do with them (he was called unexpectedly out of town that week).
Mike Gottlieb
Editor, The Feynman Lectures on Physics, Definitive Edition
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"www.feynmanlectures.info"[/URL]

Last edited by a moderator: Apr 25, 2017
5. Jun 1, 2010

### QED-Kasper

Thanks codelieb. I have to add though that I misread the text. In it Sands only mentions the expected distance. Which is also known as the MAD (mean average deviation) in statistics. The "square-root of N rule" applies to the RMS (root mean square) distance, aka standard deviation. This is what is actually being described in the text.