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Richa Nandra, Lovely Professional University Unit 20: Control Limit Theorem

Unit 20: Control Limit Theorem Notes

CONTENTS

Objectives
Introduction
20.1 Central Limit Theorem
20.2 Summary

20.3 Keywords
20.4 Self Assessment
20.5 Review Questions
20.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Define the central limit theorem

 Describe control limit theorem
Introduction

In Binomial distribution with parameters n and p is shown to be approximable by a Poisson
distribution whenever n is large and p is such that np is a constant A z 0. An important limit
theorem, known as the central limit theorem, is studied in Section 14.4. Central limit theorem
essentially states that whatever the original distribution is (as long as it has finite variance), the
sample mean computed from the observations following that distribution has an approximate
normal distribution as long as the sample size (number of observations) is large. An important
special case of this result is that binomial distribution can be approximated by an appropriate
normal distribution for large samples.

20.1 Central Limit Theorem

The Central Limit Theorem (CLT) is one of the most important and useful results in probability
theory. We have already seen that the sum of a finite number of independent normal random
variables is normally distributed. However the sum of a finite number of independent non-
normal random variables need not be normally distributed. Even then, according to the central
limit theorem, the sum of a large number of independent random variables has a distribution
that is approximately normal under general conditions. The CLT provides a simple method of
computing the probabilities for the sum of independent random variables approximately. This
theorem also suggests the reasoning behind why most of the data observed in practice leads to
bell-shaped curves.

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