Guide to Exercise 1

Now we have our Exercise 1! The Problem 1 is a simple warm up to do some Gaussian integrals and related Fourier transform. One of the important theorem in statistics is the central limit theorem. The Problem 2 will guide you to get a feeling about what that is by doing a special example that one can tract how cetral limit theorem emerges gradually. After the problem, one should be able to carry out the analytic/ numerical calculation exactly. Also, one can notice how fast the result approaches the conclusion of central limit theorem. At the same time, one can also notice the limitation of this brute force approach. Therefore, we design Problem 3 where we can carry out the “proof” of the central limit theorem. Here, one should focus on the requirements to apply the central limit theorem and its conclusion. After that, in Problem 4, we apply the central limit theorem in a simplified model of polymer physics where one can develop the connection between a random walk problem to polymer physics.