Application of Three Probability Distributions to Justify Central Limit Theorem

Abstract

This paper focused on the use of three probability distributions to justify the central limit theorem (CLT). The aim was to use the moment generating function (MGF) to prove (CLT) and also to portray the shape of different sample sizes (15, 30 and 100) of distributions of sample means on a histogram. The population distributions studied were: Normal, Gamma and Exponential distributions. In addition, sampling distribution of the mean table was constructed for better understanding of CLT. The study used simulation to simulate population distribution of Normal, Gamma and Exponential. Five hundred (500) distributions of sample means were drawn from each of the simulated population distributions at three different sample sizes (n): 15, 30 and 100. The shape of the simulated population distribution and sampling distribution of mean were presented on a histogram. The mean and standard deviation of each population distribution together with distribution of sample means at different sample sizes were also presented on the histogram plotted. The findings showed that under normal distribution, the sampling distribution of mean produced a shape like normal distribution irrespective of the sample size. Conversely, the shape of sampling distribution of mean under non-normal distributions gradually converges to normal distribution as sample size tends to infinity, while the variability of each sampling distribution decreases as the sample size increases. Therefore, CLT holds for large sample size (n ≥ 30).