Random Variable - Random Signals and Processes Primer with MATLAB - page 106

From Fig. 2.48 , we have:

f X ða xÞ¼f X ða þ xÞ: (2.367)

The mean value can be expressed as:

1 1 1

m X ¼ xf X ðxÞ d x ¼ ðx aÞf X ðxÞ d x þ af X ðxÞ d x: (2.368)

1 1 1

Keeping in mind that the second integral in ( 2.368 ) is equal to a , we arrive at:

1 1

m X ¼ xf X ða xÞ d x þ xf X ða þ xÞ d x þ a ¼ a: (2.369)

a a

Using ( 2.367 )to( 2.369 ), it follows that m X ¼ a , i.e., the mean value is equal to
the symmetry point a .
The value of r.v. for which the PDF has maximum, is called the mode . Generally,
the mean value is not equal to the mode. If the density is symmetrical, then the
mode is the point of symmetry, and thus corresponds to the mean value (see
Fig. 2.49a ). Otherwise, the mode and the mean value are different, as shown in
Fig. 2.49b .

Density and Third Central Moment

The third central moment, m 3 , is a measure of the symmetry of the random variable
and is also called a skew of the density function. If the density has symmetry around

Fig. 2.49 Mean value, modal point, and probability density. ( a ) Mean value. ( b ) Modal point