A Modified Review and Proof of Central Limit Theorem in Relation with Law of Large Numbers

Casmir Chidiebere Onyeneke 1*

1 Department of Mathematics and Statistics, Hezekiah University, Umudi, Nigeria

Abstract

The study of the central limit theorem and law of large are paramount to mathematicians and statisticians because of the undisputable role they play in probability, estimation, decision making and analysis of confidence intervals. As the fundamental theorems underlying applied mathematics, it can never be overemphasized that the distribution of the sum of a large number of independent and identically distributed approximates to normal irrespective of the nature of the underlying distribution. This is the sole theorem of the study of central limit theorem and law of large numbers. The essence of this modified review and study of these powerful keys of mathematics is to elaborate the essential proofs and applications that uphold their position. Indeed, it is hard to overstate the importance of the central limit theorem due to the fact that it is the major reason for many statistical and mathematical procedures to work. When the distribution of statistical cumulative functions is viewed in the cause of this study, they all approximate to normal under repeated observation in the same condition for a long period of occurrences. The first thing to do in every given distribution is to standardize the function if it assumes non-normal. Next, is to normalize them by the assumptions of central limit theorem and law of large numbers. In this work, different approaches are used to tackle different distributions in order to get their approximation to normal by the application of central limit theorem and law of large numbers.

Keywords

Central Limit Theorem, Law of Large Number, Discrete Random Variables, Cumulative, Moments, Probability Magnitude, Normal Distribution, Variance

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