This site demonstrates a one-dimensional random walk and relates it to other phenomena which random walks are supposed to model. The user is prompted to experiment by flipping a coin to simulate movement left or right on a number line. The supporting java applet allows the user to select the number of steps in a walk, shows those random steps on a number line, and plots the average mean squared distance for each step with a corresponding ?best fit? line. The number of walks can be increased significantly with the plots reconstructed after each iteration. There is also a 2 dimensional random walk model with the same setup at http://polymer.bu.edu/java/java/2drw/ . The reader should first read the 1d model to ?get into the spirit?.
Type of Material:
Simulation
Technical Requirements:
Java Applet should run on any platform.
Identify Major Learning Goals:
The student should be able to set up an experiment and then run the applet to ease the burden of data tracking. Questions are posed to the learner at the bottom of the page.
Target Student Population:
Undergraduate mathematics majors
Prerequisite Knowledge or Skills:
A basic understanding of probability theory.
Content Quality
Rating:
Strengths:
The idea of a random walk is described in relation to the motion of molecules and described in simple probabilistic terms. Since a large number of trials are necessary to acquire the desired result, the averages and resulting ?best fit? line would be difficult to create without the simulation.
Concerns:
The notion of mean squared distance is introduced somewhat ?out of the blue?. It would be helpful if the discussions of the ?average position? and ?spread of the positions? were two separate paragraphs, emphasizing them as two individual and related concepts for the student user. When stepping through a walk with a specific number of steps, the last step is not shown, but instead a new walk is begun. The graph indicates that a last step was taken and the mean square distance calculated.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
The applet will provide first exposure to the idea of a random walk in one dimension. Thought provoking questions are asked at the end. The user is allowed to explore and prompted to develop a conclusion regarding the relationship between the number of steps and the mean square distance by changing the number of steps and by stepping through the walk. Feedback is immediate.
Concerns:
The focus is narrow. References that put the idea into a much larger context are given, but require significant background to use. The questions may be beyond the casual user; the relationship between mean squared distance and number of steps is a linear one, but the user may have difficulty making a prediction about a physical model without further direction.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
The applet itself is extremely easy to use. Documentation is effective in developing the concept of a one-dimensional random walk prior to experimenting with the applet.
Concerns:
It would be nice to also have a plot of the distribution of the final positions.
Creative Commons:
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