Random Variable - Random Signals and Processes Primer with MATLAB - page 99

From ( 2.330 ), it follows that the zero central moment is equal to 1, while the first
central moment is equal to zero

m 0 ¼ 1 ;

(2.331)

m 1 ¼ 0 :

We want to represent the average spread of the random variable in positive as
well in negative directions from the mean value. One possible way to measure this
variation would be to consider the quantity Ef X EfXg j jg . However, it turns out to

be useless to deal with this quantity, as shown below,

EX EfXg f g for X EfXg;

EX f j EfXg j g ¼ (2.332)

E ðX EfXgÞ f g for X EfXg:

Note that both expressions on the right side of ( 2.332 ) lead to a zero value. As a
consequence, an easier way to describe the variations of the random variable around
its expected value is a variance, presented below.

2.8.2.1 Variance

The most important central moment is the second central moment , m 2 , because it
presents the average spread of the random variable in positive as well in negative
directions from its mean value. It has its own name ( variance ) and denotation, s 2 or
VAR( X ). When one would like to emphasize that s 2 is the variance of the random
variable X , one writes s X rather than s 2 .

n o

:

2

m 2 ¼ s X ¼ VAR ðXÞ¼E

ðX EfXgÞ (2.333)

From ( 2.273 ), we get the following expression for the variance:

1

2

s X 2 ¼ ðx EfXgÞ f X ðxÞ d x: (2.334)

1

This expression can be used for both continuous and discrete random variables
as explained in Sect. 2.7.4 . However, we can also give the corresponding expression
for the discrete variable X :

¼ 1
i¼1

s X 2

2

ðx i EfXgÞ PðX ¼ x i Þ: (2.335)