Page 103 - Fundamentals of Probability and Statistics for Engineers

86 Fundamentals of Probability and Statistics for Engineers

4.2 CHEBYSHEV INEQUALITY

In the discussion of expectations and moments, there are two aspects to be
considered in applications. The first is that of calculating moments of various
orders of a random variable knowing its distribution, and the second is con-
cerned with making statements about the behavior of a random variable when
only some of its moments are available. The second aspect arises in numerous
practical situations in which available information leads only to estimates of
some simple moments of a random variable.
The knowledge of mean and variance of a random variable, although very
useful, is not sufficient to determine its distribution and therefore does not
permit us to give answers to such questions as ‘What is P X 5)?' However, as
is shown in Theorem 4.1, it is possible to establish some probability bounds
knowing only the mean and variance.

Theorem 4.1: the Chebyshev inequality states that
1
P jX m X j k X † ; 4:17†
k 2

for any k > 0.
Proof: from the definition we have

Z 1 Z
2
2
2
ˆ x m X † f x†dx x m X † f x†dx
X
X
X
1 jx m X j k X
Z
2 2
k X f x†dx
X
jx m X j k X
2 2
ˆ k P jX m X j k X †:
X
Expression (4.17) follows. The proof is similar when X is discrete
Example 4.9. In Example 4.7, for three-foot tape measures, we can write
1
P jX 3j 0:03k† :
k 2
If k ˆ 2,

1
P jX 3j 0:06† ;
4

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