Random Variable - Random Signals and Processes Primer with MATLAB - page 109

The second moment is equal to:

1

x 2 l e lx d x ¼ l Gð 3 Þ
l 3

2 2

l 2 :

!
l 2 ¼

m 2 ¼ ¼ (2.378)

0

From ( 2.377 ) and ( 2.378 ), using ( 2.337 ), the variance is:

2
l 2

1
l 2 ¼

1
l 2 :

s 2 ¼ m 2 m 1 ¼ (2.379)

The third central moment is:

1

3

l e lx d x:

m 3 ¼ ðx 1 =lÞ (2.380)

0

After brief calculations, we arrive at:

=l 3

m 3 ¼ 2 : (2.381)

The nonzero third central moment is the result of asymmetry in the PDF.
Similarly, we get the fourth central moment:

1

4

=l 4

l e lx d x ¼ 9

m 4 ¼ ðx 1 =lÞ : (2.382)

0

From ( 2.372 ) and ( 2.379 )-( 2.381 ), the coefficient of skewness is given by:

3

m 3
s 3 ¼

EfðX mÞ g

c s ¼ ¼ 2 : (2.383)

s 3

The positive value ( 2.383 ) shows that the density function is skewed to the right,
as it is shown in Fig. 2.51 .
Similarly, from ( 2.373 ), ( 2.379 ), and ( 2.382 ), the coefficients of kurtosis is
given as:

4

m 4
s 4 ¼

EfðX mÞ g

c k ¼ ¼ 9 : (2.384)

s 4

The large coefficient of Kurtosis indicates that the density has a sharp peak near
its mean. This is confirmed in Fig. 2.51 .
Higher values of the moments do not have a similar physical interpretation, as do
the first four moments.