Introduction

The central limit theorem states that the average of a sum of N random variables tends to a Gaussian distribution as N approaches infinity. The only requirement is that the variance of the probability distribution for the random variables be finite.

To be specific, consider a continuous random variable x with probability density f(x). That is, f(x)Δx is the probability that x has a value between x and x + Δx. The mean value of x is defined as

<x> = ∫x f(x) dx.

Similarly the mean value of x^{2} is given by

<x^{2}> = ∫x^{2}f(x) dx.

The variance σ_{x}^{2} of f(x) is

σ_{x}^{2} = <x^{2}> - <x>^{2}.

Now consider the sum y_{N} of N values of x:

y = y_{N} = (1/N)(x_{1} + x_{2} + … + x_{N}).

We generate the N values of x from the probability density f(x) and determine the sum y. The quantity y is an example of a *random additive process*. We know that the values of y will not be identical, but will be distributed according to a probability density p(y), where p(y)Δy is the probability that the value of y is in the range y to y + Δy. The main question of interest is what is the form of the probability density f(y)?

As we will find by doing the simulation, the form of p(y) is *universal* if σ_{x} is finite and N is sufficiently large.

Method

- Generate N random variables x
_{i}that satisfy a given probability density f(x), sum them, and divide by N. - Repeat step 1 many times.
- Plot the histogram of the values of the sum y.

Problems

- First consider the uniform distribution f(x) = 1 in the interval [0,1]. Calculate <x> and σ
_{x}. - Use the default value of N and describe the qualitative form of p(y). Does the qualitative form of p(y) change as the number of measurements of y is increased for a given value of N?
- What is the approximate width of p(y) for N = 12? Describe the changes, if any, of the form and width of p(y) as N is increased. Increase N by at least a factor of 4.
- To determine the generality of your results, consider the probability density f(x) = 2e
^{-2x}for x ≥ 0. Verify that f(x) is properly normalized. (We have chosen f(x) so that its mean is the same as the mean for the uniform distribution in Problem 1.) - Consider the Lorentz distribution
f(x) = (1/π)(1/(x

^{2}+ 1),where -∞ ≤ x ≤ ∞. Use symmetry arguments to show that <x> = 0. What is the variance σ

_{x}? Do you obtain a Gaussian distribution for this case? If not, why not? - Each value of y can be considered to be a measurement. The
*sample variance*s^{2}is a measure of the square of the difference in the result of each measurement and is given by^{2}is that to compute it, we need to use the N values of x to compute the mean of y, and thus, loosely speaking, we have only N - 1 independent values of x remaining to calculate s^{2}. Show that if N >> 1, then s ≅ σ_{y}, where the standard deviation σ_{y}is given byσ

_{y}^{2}= <y^{2}> - <y>^{2}. - The quantity s is known as the
*standard deviation of the means*. That is, s gives a measure of how much variation we expect to find if we make repeated measurements of y. How does the value of s compare to your estimated width of the probability density p(y)?

References

- H. Gould, J. Tobochnik, and Wolfgang Christian,
*An Introduction to Computer Simulation Methods*(Addison-Wesley, 2006), 3rd ed., pp. 213-214.

Java Classes

- CentralApp

Updated 28 December 2009.