RandomWalk - previous index next The One-Dimensional Random...

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previous index next The One-Dimensional Random Walk Michael Fowler, UVa Physics 6/8/07 Flip a Coin, Take a Step The one-dimensional random walk is constructed as follows : You walk along a line, each pace being the same length. Before each step, you flip a coin. If it’s heads, you take one step forward. If it’s tails, you take one step back. The coin is unbiased, so the chances of heads or tails are equal. The problem is to find the probability of landing at a given spot after a given number of steps, and, in particular, to find how far away you are on average from where you started. Why do we care about this game? The random walk is central to statistical physics . It is essential in predicting how fast one gas will diffuse into another, how fast heat will spread in a solid, how big fluctuations in pressure will be in a small container, and many other statistical phenomena. Einstein used the random walk to find the size of atoms from the Brownian motion . The Probability of Landing at a Particular Place after n Steps Let’s begin with walks of a few steps, each of unit length, and look for a pattern. We define the probability function f N ( n ) as the probability that in a walk of N steps of unit length, randomly forward or backward along the line, beginning at 0, we end at point n . Since we have to end up somewhere, the sum of these probabilities over n must equal 1. We will only list nonzero probabilities. For a walk of no steps, f 0 (0) = 1. For a walk of one step, f 1 (–1) = ½, f 1 (1) = ½. For a walk of two steps, f 2 (–2) = ¼, f 2 (0) = ½, f 2 (2) = ¼. It is perhaps helpful in figuring the probabilities to enumerate the coin flip sequences leading to a particular spot.
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2 For a three-step walk, HHH will land at 3, HHT, HTH and THH will land at 1, and for the negative numbers just reverse H and T. There are a total of 2 3 = 8 different three-step walks, so the probabilities of the different landing spots are: f 3 (–3) = 1/8 (just one walk), f 3 (–1) = 3/8 (three possible walks), f 3 (1) = 3/8, f 3 (3) = 1/8. For a four-step walk, each configuration of H’s and T’s has a probability of (½) 4 = 1/16. So f 4 (4) = 1/16, since only one walk—HHHH—gets us there. f 4 (2) = ¼; four different walks, HHHT, HHTH, HTHH and THHH, end at 2. f 4 (0) = 3/8, from HHTT, HTHT, THHT, THTH, TTHH and HTTH. Probabilities and Pascal’s Triangle If we factor out the 1/2 N , there is a pattern in these probabilities: n –5 –4 –3 –2 –1 0 1 2 3 4 5 f 0 ( n ) 1 2 f 1 ( n ) 1 1 2 2 f 2 ( n ) 1 2 1 2 3 f 3 ( n ) 1 3 3 1 2 4 f 4 ( n ) 1 4 6 4 1 2 5 f 5 ( n ) 1 5 10 10 5 1 This is Pascal’s Triangle —every entry is the sum of the two diagonally above. These numbers are in fact the coefficients that appear in the binomial expansion of ( a + b ) N . For example, the row for 2 5 f 5 ( n ) mirrors the binomial coefficients: () 5 5 4 32 23 4 5 51 0 1 0 5 ab a a b a b a b a b b + = + +++ + .
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This note was uploaded on 12/07/2011 for the course PHYSICS 152 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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RandomWalk - previous index next The One-Dimensional Random...

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