Digital Signal Processing Reference
In-Depth Information
2
=1
NORMAL PDF: m=0,
σ
0.4
0.35
0.3
0.25
0.2
d
0.15
0.1
0.05
x4
x1
x2
x3
0
−
4
−
3
−
2
−
1
0
x
1
2
3
4
Fig. 4.1
Normal PDF
Using integral 4 in Appendix
A
, we have:
1
p
4
¼ b
2
p
4
b
2
4
b
2
p
s
2
u
2
e
u
2
d
u ¼
p
p
¼
:
(4.9)
0
Therefore, the constant
b
is the standard deviation of the normal random
variable.
Using the results obtained in (
4.5
) and (
4.9
), the PDF (
4.1
) can be rewritten in its
more common form as:
1
2
p
e
ð
x
m
Þ
2
p
f
X
ðxÞ¼
;
1<x<1
(4.10)
2
s
2
s
As a consequence, the normal PDF is completely determined by its two first
moments and, consequently, it is often done in the simple form:
f
X
ðxÞ¼Nðm; s
2
Þ:
(4.11)
The density is shown in Fig.
4.1
for
m ¼
0 and
s
2
¼
1.
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