Page 262 - DMTH404_STATISTICS

Statistics Richa Nandra, Lovely Professional University

Notes Unit 18: The Weak Law

CONTENTS

Objectives
Introduction
18.1 Summary
18.2 Keywords

18.3 Self Assessment
18.4 Review Questions
18.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the weak laws
 Describe some examples related to weak law

Introduction

James Bernoulli proved the weak law of large numbers (WLLN)around 1700 which was published
posthumously in 1713 in his treatise Ars Conjectandi. Poisson generalized Bernoulli’s theorem
around 1800, and in 1866 Tchebychev discovered the method bearinghis name. Later on one of
his students, Markov observed that Tchebychev’s reasoning can be used to extend Bernoulli’s
theoremto dependent random variables as well.
In 1909 the French mathematician Emile Borel proved adeeper theorem known as the strong law
of large numbers that furthergeneralizes Bernoulli’s theorem. In 1926 Kolmogorov derived
conditions that were necessary and sufficient for a set of mutually independent random variables
to obey the law of large numbers.

18.1 Weak Law of Number

Let X be independent, identically distributed Bernoulli randomVariables such that
i
P(X ) = p, P(X = 0) = 1 – p = q,
i i
and let k = X + X + ... + X represent the number of “successes”in n trials. Then the weak law due
1 2 n
to Bernoulli states that [see Theorem 3-1, page 58, Text]

 k  pq
P   p    2 ...(18.1)

 h  n
i.e., the ratio “total number of successes to the total numberof trials” tends to p in probability as
nincreases.

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