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...the CLT as it applies to probability and statistics. We will give its history as well as clear understanding of its power. In addition to reproducing previous work[1], we will provide the MATLAB code used to perform further demonstrations. Our program will select 30 integers between one and six, as in Lazari et. al. It will then compute each individual mean (L1) and store it in a list (L5) while repeating itself n times, where n is the total number of ensembles. Upon completion, distribution plots are obtained for the n means as well as a combined histogram for each individual (L5). For a very large n, the program does indeed demonstrate that the distribution of the sample means is really normal as in Lazari et al.
Key words: Matlab, Central Limit Theorem.
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INTRODUCTION
The Central Limit Theorem (CLT) states that for random samples taken from a population with a standard deviation of s (variance [s.sup.2]), that is not necessarily normal (having unique values of u and s, respectively), the sampling distribution of the sample means are approximately normal when the sample size is large enough (n[greater than or equal to]35); having a mean ([u.sub.x]) and a standard deviation s/n. In describing this relationship between sample mean and population samples, a more in depth way of stating the theorem is: The sample means of n, independent and identically distributed random variables approaches the normal distribution as n increases.
The normal distribution function originates from what is known as the probability density function (Gaussian function):
f(x) = [1/[[sigma][square root of (2[pi])]]] [e.sup.-(x-[mu])[.sup.2]/2[[sigma].sup.2]] - x < x < [infinity] (1)
This form of the density function is used as the basic format and can be transformed depending upon the actual mean (m) and standard deviation ([sigma]) obtained from the random samples. The graph associated with the density function is known as the bell shaped curve as seen in Fig. 1.
[FIGURE...
NOTE: All illustrations and photos
have been removed from this article.
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