Page 158 - DMTH404_STATISTICS

Statistics Richa Nandra, Lovely Professional University

Notes Unit 11: Chebyshev’s Inequality

CONTENTS

Objectives
Introduction
11.1 Chebyshev’s Inequality
11.2 Summary

11.3 Keywords
11.4 Self Assessment
11..5 Review Questions
11.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Apply chebyshev’s inequality

 Give example of chebyshev’s inequality
Introduction

We have discussed different methods for obtaining distribution functions of random variables
or random vectors. Even though it is possible to derive these distributions explicity in closed
form in some special situations, in general, this is not the case. Computation of the probabilities,
even when the probability distribution functions are known, is cumbersome at times. For
instance, it is easy to write down the exact probabilities for a binomial distribution with
1
parameters n = 1000 and p = . However computing the individual probabilities involve
50
factorials for integers of large order which are impossible to handle even with speed computing
facilities.

In this unit, we discuss limit theorems which describe the behaviour of some distributions when
the sample size n is large. The limiting distributions can be used for computation of the
probabilities approximately.
Chebyshev’s inequality is discussed, as an application, weak law of large numbers is derived
(which describes the behaviour of the sample mean as n increases).
11.1 Chebyshev’s Inequality

We prove in this section an important result known as Chebyshev’s inequality. This inequality
is due to the nineteenth century Russian mathematician P.L. Chebyshev.

We shall begin with a theorem.

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