Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 38

m X ¼Eð X Þ¼ Z

1

xp X ð x Þ dx ð 1 : 26 Þ

1

and it represents the center around which the values of X are expected to be
distributed.

1.3.2.8 The Second Moment

The second moment of a random variable X is defined as

X ¼E X 2 ¼ Z

1

m ð 2 Þ x 2 p X ð x Þ dx ð 1 : 27 Þ

1

and it represents the expected value of the square of the deviations of a random
variable X from its mean value m X .

1.3.2.9 The Variance

The second central moment (or variance) of a random variable is defined as

o ¼ Z

1

n

r 2 ¼ var ð X Þ¼E ð X m X Þ 2 Þ 2 p X ð x Þ dx : ð x m X

1

The quantity r X ¼ var ð X p is called the standard deviation of X. The variance

indicates how far the values of X are spread around the mean. Hence, the vari-
ance gives a measure of the randomness of a random signal. The quantity
r X ¼

p

var ð X Þ is called the standard deviation of X.

Note:

¼E X 2 2m X Eð X Þþ m X ¼E X 2 m X

r X ¼E X 2 2m X þ m X

ð 1 : 28 Þ

1.3.2.10 The Gaussian pdf

The Gaussian pdf is an important probability density function which is often
encountered in real-world applications. A random variable X is said to be Gaussian
if its pdf is given by:

p ð x Þ¼ 1
r

ð x m Þ 2
2r 2

p e

2p