Random Variable - Random Signals and Processes Primer with MATLAB - page 102

yielding

1
12 0 ¼

1
12 :

s 2

X 1 ¼

(2.346)

Similarly, for the r.v. X 2 , we get:

ð ð

2 2

3
2 ;

1
3 8 1

7
3 ;

x 2 d x ¼

m 1 ¼ x d x ¼ m 2 ¼ ½ ¼ (2.347)

1 1

resulting in:

7
3

9
4 ¼

1
12 :

s 2

X 2 ¼

(2.348)

Note that the random variables X 1 and X 2 have equal variance given that both
variables have the same uniform range.
Generally speaking, the variance of the uniform random variable with a range
width D can be presented as:

2
12 :

¼ D s 2 (2.349)

2.8.2.2 Properties of Variance

The variance of a random variable X has the useful properties described in the
continuation.

P.1 Adding a constant to the random variable does not affect its variance:

VAR ðX bÞ¼ VAR ðXÞ , for any constant b .

From ( 2.333 ), we write:

n o

2

VAR ðX bÞ¼E ðX b EfX bgÞ : (2.350)

According to ( 2.244 ),

EfX bg¼EfXgb; (2.351)

resulting in:

2

VAR ðX bÞ¼EfðX EfXgÞ g¼ VAR ðXÞ: (2.352)