Digital Signal Processing Reference
In-Depth Information
Exercise 5.8
Find the PDF of the random variable
Y
if
a
for
X a
Y ¼
(5.234)
a
for
X > a
The random variable
X
has a PDF of
f
X
ðxÞ¼
e
x
;
x
0
(5.235)
Answer
PfY ¼ ag¼PfX ag¼
1
e
a
;
(5.236)
PfY ¼ag¼PfX>ag¼
1
PfX ag¼
1
ð
1
e
a
Þ¼
e
a
:
(5.237)
The random variable
Y
is discrete and has two discrete values as given in (
5.236
)
and (
5.237
).
f
Y
ðyÞ¼ð
1
e
a
Þdðy aÞþ
e
a
dðy þ aÞ:
(5.238)
Exercise 5.9
A random variable
Y
, given an exponential random variable
X
with a
parameter
l
, has the following conditional PDF:
x
e
xy
for
y
0
;
f
Y
ðyjxÞ¼
(5.239)
0
otherwise
:
Find the joint density of the variables
X
and
Y
.
Answer
A conditional PDF can be expressed in terms of the joint PDF and PDF of
the variable
X
:
f
YX
ð
y
;
x
Þ
f
X
ðxÞ
f
Y
ðyjxÞ¼
:
(5.240a)
From here,
f
YX
ðy; xÞ¼f
Y
ðyjxÞf
X
ðxÞ¼l
e
lx
x
e
xy
¼ lx
e
xðlþyÞ
;
for
x
0
;
y
0
:
(5.240b)
f
XY
ðx; yÞ¼
0
;
for
x<
0
;
y<
0
:
(5.240c)
Exercise 5.10
For a given Poisson flow of events with an intensity of
l
, form a new
flow of events by placing a new event in the middle of the two successive Poisson
events. For the random variable
Y
, which describes the time between two successive
events in a new flow of events, find the PDF.
Search WWH ::
Custom Search