Montgomery College Rockville Campus Central Theorem Limit Matlab Task

1 I

N

TRODUCTION

One of the most important theorems in the study of probability and random processes is the Central Limit Theorem

(CLT). Let us define the sample sum of independent random variables, ,where each random variable,

, has a finite mean, , and a finite variance, . The Central Limit Theorem states that in this case, has a

probability density function (pdf) that approaches a Gaussian pdf with mean and a variance

as grows large (but not infinite!).

Put it this way: if we add a bunch of well-behaved independent random variables together, the pdf of the sum is

approximately Gaussian, and the approximation gets better as we add more terms (i.e., more independent random

variables) to the sum. The approach to Gaussian in the limit as N gets large does not depend on the shape of the

pdfs, or even on the random variables being identically distributed. All that is required is that they are

independent!

We will study the CLT during the course of this project, but you do not need to wait for us to cover it in class. You

can do all of the project without us discussing the CLT in class.

Warning #1: Thinking is required!

Warning #

2

: Follow directions!

Warning #3: It isn’t on the web, so don’t bother looking.

2 PROJECT TASKS

Perform the following tasks, then document your results and submit them in written form in accordance with the

instructions in Section 3, below. You may (and should) use this document as a template for your report.

2.1 Sum of Independent, Identically Distributed (iid) Random Variables from U(0,1)

Generate the sum of N random variables distributed for .

MATLAB users will use the function rand.

Generate a large number of such sums, say 100,000 or more, for each value of . Plot a histogram of the results

for each , scaling the histogram appropriately to be a probability density function. (Note that the random variable

upon which the histogram is based is the sum of N random variables, and there are a large number of such sums in

each experiment. See the MATLAB skeleton for some hints.) In each case, compute the mean and standard

N Y =

Xk

k=1

N

∑

Xk mk

2

s k Y

m = mk

k=1N

∑ σ2 = σ k

2k=1N∑NXk

Xk ~ U(0,1),k = 1,2…,N NNN === 2, 6, 12

NN