Random Variable - Random Signals and Processes Primer with MATLAB - page 119

Fig. 2.53 Illustration of Chebyshev theorem (first form)

We introduce the zero mean variable shown in Fig. 2.53b (see Sect. 2.8.2.3 )

Y ¼ X m X ;

(2.433)

resulting in:

ð 1

e

s 2 y 2 f Y ðyÞ d yþ y 2 f Y ðyÞ d y:

X

(2.434)

1 e

Replacing y by value ðeÞ in ( 2.434 ) makes this inequality even stronger:

2 3

ð

e

1

e 2 f Y ðyÞ d y; ¼ e 2 ð

e

1

4 5 ;

s X e 2 f Y ðyÞ d y þ

f Y ðyÞ d y þ f Y ðyÞ d y

1 e 1 e

¼ e 2 PfjYjeg¼e 2 PfjX m X jeg:

(2.435)

Finally, from ( 2.435 ), we have:

PfjX m X jegs 2

X =e 2

: (2.436)

2.8.5.2 Second Form of Inequality

Replacing e ¼ ks X in ( 2.430 ), we can obtain the alternative form of Chebyshev

inequality

=k 2 PfjX m X jks X g 1 : (2.437)

The inequality ( 2.437 ) imposes a limit of 1/ k 2 , where k is a constant, to the
probability at the left side of ( 2.437 ).