Random Variable - Random Signals and Processes Primer with MATLAB - page 97

The most important moments are the first and the second.
The first moment is the mean value of the r.v. X ,

m 1 ¼ EfX 1

g¼EfXg: (2.319)

The second moment is called the mean squared value and is a measure of the
strength of the random variable

m 2 ¼ EfX 2 g: (2.320)

The square root of the mean squared value

p

EfX 2 g is called the rms (root mean

squared) value of X

rmsðXÞ¼

p

EfX 2 g : (2.321)

The name “moments” comes from mechanics. If we think of f X ( x ) as a mass
distributed along the x -axis [THO71, p. 83], [CHI97, p. 94], then:

The first moment calculates the center of gravity of the mass.

The second moment is the central moment
of inertia of the distribution of mass around the center of gravity. (2.322a)

Next, we interpret moments of X which represents a random voltage across a 1 O

resistor. Then:
m 1 ¼ EfXg represents the DC (direct-current) component,

m 2 ¼ EfX 2 g represents the total power

(2.322b)

ð including DC Þ:

Example 2.8.1 We want to find the first three moments of the discrete random
variable X with values 0 and 1 and the corresponding probabilities P {0} ¼
P {1} ¼ 0.5.

Solution The random variable X is a discrete variable and its moments can be
calculated using ( 2.317 ) as shown in the following.
The first moment:

m 1 ¼ 0 0 :

5 þ 1 0

:

5 ¼ 0

: :

5 (2.323)

The second moment:

5 þ 1 2

m 2 ¼ 0 0 : 0 : 5 ¼ 0 : 5 : (2.324)

Third moment:

5 þ 1 3

m 3 ¼ 0 0 : 0 : 5 ¼ 0 : 5 (2.325)