Digital Signal Processing Reference
In-Depth Information
Fig. 2.63
(
a
) PDF and (
b
) distribution of the variable
Y
a
f
x
(x)
b
Y
0.5
U
x
X
−
U
Fig. 2.64
Input random variable and its transformation. (
a
) PDF. (
b
) Transformation
Find the PDF, distribution, and variance of the random variable
Y
, when the
random variables
X
and
Y
are related, as shown in Fig.
2.64b
.
Answer
The output random variable
Y
is discrete, taking the value
U
when the
variable
X
is positive and the value
U
when the variable
X
is negative.
Therefore,
PfY ¼ Ug¼PfX>
0
g¼
0
:
;
PfY ¼Ug¼PfX<
0
g¼
0
:
:
5
5
(2.484)
The corresponding PDF will have delta functions at the discrete points
U
and
U
, each with the area equal to 1/2 (Fig.
2.65a
). The distribution will have two
jumps, each equal to 0.5 at the discrete points
U
and
U
, as shown in Fig.
2.65b
.
Note that
the PDF is symmetrical around
y ¼
0, resulting in
E
{
Y
}
¼
0.
The corresponding variance is:
2
2
s
Y
¼ EfYg
ðEfYgÞ
2
5
U
2
¼ U
2
¼
0
:
þ
0
:
5
ðUÞ
:
(2.485)
Search WWH ::
Custom Search