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). Read more

This work investigates the rate of return from two portfolio management strategies. We first examine the return from total investment which includes both investment in the risky stock and investment in the risk-free asset. Secondly, we examine the return from investment in the risky stock only. We derive some optimality properties for the two portfolio management strategies. We show that the limiting behaviour of the rate of return on total investment is determined by the limiting behaviour of a related diffusion process. Read more

The gamma distribution is one of the continuous distributions; the distributions are very versatile and give useful presentations of many physical situations. They are perhaps the most applied statistical distribution in the area of reliability. In this paper, we present the study of properties and applications of gamma distribution to real life situations such as fitting the gamma distribution into data, burn-out time of electrical devices and reliability theory. The study employs the moment generating function approach and the special case of gamma distribution to show that the gamma distribution is a legitimate continuous probability distribution showing its characteristics. Read more

Ridge regression as a solution to multicollinearity depends on the value of k, the ridge biasing constant. Since no optimum value can be found for k, as k is generally bounded between 0 and 1, i.e. 0≤ k ≤1 and it varies from one application to another. This has posed a major limitation of ridge regression in that ordinary inference procedures are not applicable and exact distributional properties are not known; and the choice of the biasing constant, k, is a judgmental one. This work examined the effect of ridge biasing constant, k, on different sample sizes using data combination from gamma and exponential distributions when multicollinearity exists. The sample sizes of 140, 100, 80, 50, 30, 20 and 10 and ridge constants, k=0.01, 0.02, . . . ., 0.1 respectively were used in the study. The Anderson Darling Test was used to check for the distribution of the data which were found to follow gamma and exponential distributions. The findings lay credence to how the ridge regression drastically remedies the effect of multicollinearity among independent variables. The study also revealed that the VIF consistently decreased as the ridge constant increased. While the ridge regression has a slight effect on the R-squared, sample sizes were found not to have any significant change or pattern on the VIFs. Since the study has shown that the VIF reduced drastically as the ridge constant increases, it is recommended to use a VIF that reduces multicollinearity to an acceptable minimum while maximizing the R-squared. This study recommends a ridge constant of 0.1 as all multicollinearity issues have been remedied at more than 90% if not completely. The study recommends using a large sample size to help stabilize the R2 values while remedying multicollinearity. Read more