Random Variable - Random Signals and Processes Primer with MATLAB - page 118

From here, we can easily find the variance as:

2
l 2

1
l 2 ¼

1
l 2 :

s 2 ¼ m 2 m 1 ¼ (2.429)

This is the same result as that obtained in ( 2.379 ).

2.8.5 Chebyshev Inequality

Sometimes we need to estimate a bound on the probability of how much a random
variable can deviate from its mean value. A mathematical description of this
statement is provided by Chebyshev inequality .
The Chebyshev inequality states this bounding in two different forms for r.v.
X in terms of the expected value m X and the variance s X .
The Chebyshev inequality is very crude, but is useful in situations in which we
have no knowledge of a given random variable, other than its mean value and
variance. However, if we know a density function, then the precise bounds can be
found simply calculating the probability of a desired deviation from the mean value.

2.8.5.1 First Form of Inequality

The probability that the absolute deviation of the random variable X from its
expected value m X is more than e is less than the variance s X divided by e 2 ,

g s X =e 2

PX f j m X j e : (2.430)

In the continuation, we prove the formula ( 2.430 ).
Using the definition of the variance, we can write:

1

2

s X ¼ ðx m X Þ f X ðxÞ d x

1

ð ð 1

m X e m X þe

2 2 2

¼ ðx m X Þ f X ðxÞ d x þ ðx m X Þ f X ðxÞ d x þ ðx m X Þ f X ðxÞ d x:

1 m X e m X þe

(2.431)

Omitting the middle integral (see Fig. 2.53a ), we can write:

m X e

ð 1

2 2

s X

ðx m X Þ f X ðxÞ d x þ ðx m X Þ f X ðxÞ d x: (2.432)

1 m X þe