Random Variable - Random Signals and Processes Primer with MATLAB - page 98

Note that all three moments are equal. The same is true for all other moments

5 þ 1 n m n ¼ 0 0 : 0 : 5 ¼ 0 : 5 : (2.326)

Example 2.8.2 In this example, we want to calculate the first three moments of the
uniform random variable in the interval [0, 1].
The density function is:

1 for 0 x 1 ;

f X ðxÞ¼ (2.327)

0 otherwise :

Solution The random variable X is a continuous variable and its moments are
calculated using ( 2.316 ):

ð

1

1
2 ;

m 1 ¼ x d x ¼

0

ð

1

1
3 ;

x 2 d x ¼ m 2 ¼

0

ð

1

1
4 :

x 3 d x ¼

m 3 ¼ (2.328)

0

2.8.2 Central Moments

The moments described in the previous section depend on the mean value. If, for
example, the mean value is very high, then the deterministic component of the
variable is dominant, and the variable nearly loses its “randomness”.
To overcome this situation, another class of moments, taken around the mean
value, are useful. The mean value is subtracted from the signal, and thus the
corresponding mean value does not depend on the mean value of the random
variable.
The moments around the mean value are called central moments and are defined
as expected value of the function

n

gðXÞ¼ X EfXg ð Þ : (2.329)

Central moments are denoted by m n

n

m n ¼ EfðX EfXgÞ g: (2.330)