The One-Dimensional Random Walk
Random movement is important for understanding the microscopic world in Nature, because atoms and molecules move randomly. How can we describe the random motion of molecules in, say, a gas? Molecules are too small to see, so to help us think concretely we replace a molecule with something we can see: a wandering ant. If a wandering ant starts at a lamp post and takes steps of equal length along the street, how far will it be from the lamp post after a certain number, say N, steps? Though this question is seemingly trivial, it poses one of the most basic problems in statistical science.
First, let's take a look at a simple coin-flipping experiment. Suppose you flip ten coins, count the number of heads, and plot this number on a histogram. What do you suppose the histogram would look like after, say, 100 trials? Do you believe it will look something like the histogram in Figure 1? What do you suppose will happen if we were to make 1,000 trials? 10,000 trials? What you are forming is a bell-shaped curve (also called a normal or Guassian distribution).
What does Figure 1 tell us? Randomness (coin-flipping) leads to some sort of predictable outcome (the bell-shaped curve).
This Applet relates random coin-flipping to random motion. It is
easiest to visualize random motion (random walk) along one line, that
is, in one dimension. Call
Figure 2: Diagram of a wandering ant. How many steps do you think he has already taken if his starting-point was at the ``Lamp post''?
Choose the direction of the step the ant will take by flipping
a penny: (1) If it is a head, the ant steps right and
Do this activity with a partner. Use a silver-colored coin (nickel, dime, or quarter) to represent the position of the ant. To begin, put the ``ant'' in a center cell (the position of the lamp post). The ant steps from one cell to the next, right or left randomly, depending on whether the penny comes up heads or tails, respectively.
What is so important about the graph? Science is about making predictions, often based on measuring some quantities. You will find that the measure of the mean squared distance is one of the most important and powerful measures of a random walk.
How far from the origin does the wandering ant end up after some
What does theory have to say about the value of the average position
of many random walkers? Answer: The value of the average position is
zero, the position of the starting point (the lamp post)! How can this
be? One word gives the reason: Symmetry! In this case symmetry
means that moving right is just as likely as moving left. After any
fixed number of steps, the walker is equally likely to be to the right
of the starting point (positive displacement) as to the left (negative
displacement). Moreover, for a given number of steps, the distance
Now that you've had a chance to experiment, can you answer these questions?