International Journal of Probability and Statistics 2015, 4(2): 37-41 DOI: 10.5923/j.ijps.20150402.01 Estimation of the Mean and Variance of a Univariate Normal Distribution Using Least-Squares via the Differential and Integral Techniques C. R. Kikawa * , M. Y. Shatalov, P. H. Kloppers Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, South Africa Abstract Two new approaches (method I and II) for estimating parameters of a univariate normal probability density function are proposed. We evaluate their performance using two simulated normally distributed univariate datasets and their results compared with those obtained from the maximum likelihood (ML) and the method of moments (MM) approaches on the same samples, small n = 24 and large n = 1200 datasets. The proposed methods, I and II have shown to give significantly good results that are comparable to those from the standard methods in a real practical setting. The proposed methods have performed equally well as the ML method on large samples. The major advantage of the proposed methods over the ML method is that they do not require initial approximations for the unknown parameters. We therefore propose that in the practical setting, the proposed methods be used symbiotically with the standard methods to estimate initial approximations at the appropriate step of their algorithms. Keywords Maximum likelihood, Method of moments, Normal distribution, Bootstrap samples 1. Introduction Statistical inference is largely concerned with making logical conclusions about a population using an observed section or part of the entire population referred to as the sample [1]. The reference population can always be represented using an appropriate probability framework which is usually written in terms of unknown parameters. For instance the crop yield obtained when a certain fertilizer is applied can be assumed to follow a normal distribution with mean μ, and standard deviation , ; it is thereafter required to make inferences about the parameters, μ and  using the statistics  � and  that are estimated based on the sample of crop yield and then inferences made on the total crop yield. Note that in this work we only deal with one aspect of statistical inference that is estimation and two novel approaches are discussed in this case. Let  be a single realisation from a univariate normal density function with mean μ, and standard deviation , which implies that ~N(μ, ) with −∞< μ <∞, > 0. In this paper, simple and computationally attractive methods for estimating both μ and  of a univariate normal distributionfunction are proposed. However, methods for estimating the sufficient parameters of a univariate normal density function are well known such as the method of * Corresponding author: richard.kikawa@gmail.com (C. R. Kikawa) Published online at http://journal.sapub.org/ijps Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved moments and the maximum likelihood method [2, 3], but all these are computationally intensive. Again, much as the maximum likelihood estimators have higher probability of being in the neighbourhood of the parameters to be computed, in some instances the likelihood equations are intractable in the absence of high computing gadgets like computers. Though the method of moments could quickly be computed manually by hand, its estimators are usually far from the required quantities and for small samples the estimates are often times outside the parameter space [4, 5]. In all it is not worthwhile to rely on the estimates from the method of moments. 1.1. Generalized Probability Density Function When a dataset is presented and critically observed for any characteristics that it may exhibit; statistically called exploratory data analysis, we usually want to study its pattern that can vaguely lead us to a possible probability density function (pdf) that can be taken as its probability frame-work for those data. However, if it requires one to build a whole new frame-work or model, then a lot of work has to be done which is quite demanding. In this section we present a frame-work that nearly suits all the pdfs of continuous random variables  ∧  () ∅  � − ℶ �,   ≤≤  (1.1) where   and   indicate the domain of applicability: often times from −∞to ∞ or from 0 to ∞ depending on the framework under consideration. ∅  , is the actual shape