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EE351K Probability, Statistics and Random Processes
Spring 2008
Instructor: Vishwanath
{
sriram
}
@ece.utexas.edu
Homework 8: Solutions
Problem 1
The count of students dropping the course Probability and Stochastic Processes is known
to be a Poisson process of rate 0.1 drops per day. Starting with day 0, the ﬁrst day of the semester, let
D
(
t
)
denote the number of students that have dropped after
t
days. What is
p
D
(
t
)
(
d
)
?
Solution:
D(t): Poisson process with the parameter
λ
= 0
.
1
.
p
D
(
t
)
(
d
) =
(
λt
)
d
d
!
e

λd
=
(0
.
1
t
)
d
d
!
e

0
.
1
d
.
Problem 2
For an arbitrary constant
a
, let
Y
(
t
) =
X
(
t
+
a
)
. If
X
(
t
)
is a stationary random process,
is
Y
(
t
)
stationary?
Solution:
X
(
t
)
is stationary. Thus,
F
X
(
t
1
)
,...,X
(
t
k
)
(
x
1
,...,x
k
) =
F
X
(
t
1
+
τ
)
,...,X
(
t
k
+
τ
)
(
x
1
,...,x
k
)
,
for all time shifts
τ
, all
k
, and all choices of
sample times
t
1
,...,t
k
.
Then,
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