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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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