Digital Signal Processing Reference
In-Depth Information
Fig. 3.16
Convolution of uniform PDFs
From (
3.217
), the PDF of the random variable
X
is equal to the convolution of
the PDFs of the random variables
X
1
and
X
2
. In this case, it is convenient to present
the convolution graphically, as shown in Fig.
3.16
.
3.6 Numerical Exercises
Exercise 3.1
The joint random variables
X
1
and
X
2
are defined in a circle of a
radius
r ¼
2, as shown in Fig.
3.17
. Their joint PDF is constant inside the circle.
Find and plot the joint PDF and the marginal PDFs. Determine whether or not the
random variables
X
1
and
X
2
are independent.
Answer
The area
A
in Fig.
3.17
is:
A ¼ r
2
p ¼
4
p:
(3.222)
The volume below the joint density is the height of the cylinder which, according
to (
3.37
) must be unity, is shown in Fig.
3.18
.
The joint density is:
x
1
þ x
2
4
1
=
4
p
for
;
f
X
1
X
2
ðx
1
; x
2
Þ¼
(3.223)
0
otherwise
:
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