Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 232

P.5 By increasing x , the PDF approaches to zero.

Theoretically,

lim
x!1

f X ðxÞ¼ 0 : (4.21)

However, the PDF practically approaches zero for finite values of x , as shown
in Sect. 4.2 .

4.2 Normal Distribution

4.2.1 Definition

Using the definition ( 2.10 ) and ( 4.10 ), we define the normal distribution as:

ð ð

x x

1
2 p

e ð x m Þ 2

F X ðxÞ¼PfX xg¼ f X ðxÞ d x ¼ p

d x: (4.22)

2 s 2

s

1 1

This distribution is shown in Fig. 4.4 for m ¼ 0 and s 2

¼ 1.
Next we consider how the different values of the mean value and variance affect
the shape of the normal distribution.

2 =1

NORMAL DISTRIBUTION FUNCTION m=0, σ

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
− 5 − 4 − 3 − 2 − 1 0

x

12 34 5

Fig. 4.4 Normal distribution