Binomial Distribution, Diffusion, Central Limit Theorem

The object of this problem is to see the relationship between random walks, the binomial distribution, and the normal (Gaussian) distribution.

• 1. Create NumAve random walks and record their final endpoint.

• 2. Make a histogram of the endpoints.

• 3. Compare the result with the exact result obtained by applying the binomial distribution. Nelson's book has a discussion of this as do a large number of books on introductory statistical physics, Berg Links to an external site. and, of course, the web.

• 4. Plot the results.

• 5. Compare the exact result to that obtained for a Gaussian with the correct variance. The fact that this approaches a Gaussian is an example of the "Centeral Limit Theorem".

• 6. Give an example of a biological experiment where it is reasonable to assume that a measurement is distributed as a Gaussian, and an example where you would not expect a Gaussian distribution.

Look at the hints file hw3/1d_diff/binomial_hints.py. Much of the code is already written and you need to fill it in by calculating the appropriate formulas. Plot the results and upload them.