Digital Signal Processing Reference
In-Depth Information
Chapter 4
Normal Random Variable
4.1 Normal PDF
4.1.1 Definition
The most important density function is the normal PDF which is present in almost
all science and scientific techniques. This is why the PDF is called normal, because
it presents a normal behavior for a lot of random appearances. It is also known as a
Gaussian random variable in honor of famous mathematician K.F. Gauss who used
it in his important works in the theory of probability. However, it was first defined
by the mathematician De Moivre in 1733.
The random variables which are described using a normal PDF are called
normal
or
Gaussian random variables
.
The normal PDF of the random variable
X
with a range of
x
is defined as:
2
e
ð
x
a
Þ
1
2
p
2
b
2
f
X
ðxÞ¼
p
; 1<x<1;
(4.1)
b
where
a
and
b
are constants,
b >
0, and
1<a<1
.
Below we find the meaning of the constants
a
and
b
.
2
x
e
ð
x
a
Þ
1
1
1
2
p
2
b
2
EfXg¼m
X
¼
xf
X
ðxÞ
d
x ¼
p
d
x
b
1
1
2
ðx a þ aÞ
e
ð
x
a
Þ
1
1
2
p
2
b
2
¼
p
d
x
b
1
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