Random Variable - Random Signals and Processes Primer with MATLAB - page 104

has zero mean:

EfYg ¼ EfXm X g ¼ EfXgm X ¼ 0 : (2.358)

The standardized random variable Y , derived from the r.v. X , using

X m X
s

Y ¼ (2.359)

has zero mean and variance equal to 1.
The mean value is:

EfYg ¼ E X m X
s

1
s EfXm X g ¼

1
s EfXgm X

1
s ½m X m X

¼ ½ ¼

¼ 0 : (2.360)

The variance is:

(

2 )

X m X

s E X m X

VAR fYg ¼ E : (2.361)

s

According to ( 2.360 ), the second term on the right side of ( 2.361 ) is zero,
resulting in:

(

2 )

s 2
s 2 ¼ 1

X m X
s

1
s 2 EfðXm X Þ

2

VAR fYg ¼ E ¼ g ¼ : (2.362)

2.8.3 Moments and PDF

As mentioned before, moments are often used as statistical descriptors of certain
characteristics of the random variable, which in turn is completely described using
probability density. Thus, moments may be considered as parameters of probability
density, as discussed below.
Consider the mean squared deviation of a random variable X around the constant a :

1

2

s ¼ ðxaÞ f X ðxÞ d x: (2.363)

1