Random Variable - Random Signals and Processes Primer with MATLAB - page 120

Example 2.8.14 Let X be a random variable with the mean value m X ¼ 0 and
variance s X ¼ 3. Find the largest probability that jj 2. (The PDF is not known).
Solution Using the First form ( 2.423 ), we have:

2 2

PfjXj 2 g 3 = ¼ 3 = 4 ¼ 0 : 75 : (2.438)

This result can be interpreted as follows: If the experiment is performed large
number of times, then the values of the variable X are outside the interval [ 2, 2]
approximately less than 75% of the time.
The same result is obtained using the second form ( 2.434 )

n o 1

jXjk 3

p

=k 2 P :

(2.439)

Denoting

k p

¼ 2 ; (2.440)

we get:

=k 2

1 ¼ 3 = 4 ¼ 0 : 75 ; (2.441)

which is the same result as ( 2.438 ).
Let us now consider the random variable Y with known PDF, and with the same
mean value m Y ¼ 0 and variance s Y ¼ 3. Consider that the random variable Y is the
uniform r.v. in the interval [ 3, 3], as shown in Fig. 2.54 . In this case, we can find
the exact probability that jj 2, because we know the PDF. We can easily verify
that the mean value is equal to 0 and the variance is equal to 3.

m Y ¼ 0 ;

s Y ¼ 3

: (2.442)

Fig. 2.54 Illustration of Chebyshev inequality for uniform variable