Hi i want everything solved on paper1
EENG635: Probability and Stochastic Processes
Homework
#04
: Due on Wednesday, April 08, 2015
This homework carries 5 points towards your fnal grade
1)
(5 points)
Consider a random vector,
X
=
b
X
1
X
2
X
3
···
X
n
B
T
. Let
E
(
X
i
) =
?
i
and let
?
=
b
?
1
?
2
?
3
···
?
n
B
T
. Let
C
= [
c
ij
]
1
?
i
?
n
1
?
j
?
n
, be the covariance matrix, where
c
ij
=
E
[(
X
i

?
i
)(
X
j

?
j
)]
and
c
ii
=
V ar
(
X
i
) =
?
2
i
. Let
?
= [
?
ij
]
1
?
i
?
n
1
?
j
?
n
be the correlationcoeFfcient matrix, where
?
ij
=
c
ij
?
i
?
j
and
?
ii
= 1
,
?
i
. Show that
?
is positive semidefnite.
2)
(10 points)
Consider a random vector,
X
=
b
X
1
X
2
X
3
···
X
n
B
T
. Let
E
(
X
i
) =
?
i
and let
?
=
b
?
1
?
2
?
3
···
?
n
B
T
. Let
C
= [
c
ij
]
1
?
i
?
n
1
?
j
?
n
, where
c
ij
=
E
[(
X
i

?
i
)(
X
j

?
j
)]
and
c
ii
=
V ar
(
X
i
) =
?
2
i
.
Show that
1
n
E
b
(
X

?
)
T
C

1
(
X

?
)
B
= 1
.
3) Ms. Cooper wishes to attend a conFerence which would take place in Hotel Mariott or Hotel Hilton. She is
not sure where the conFerence will take place but has to make her decision to book a room, immediately.
The rooms in Hotel Mariott cost $
180
per night and those in Hilton cost $
150
per night. The taxi charge
From Hotel Mariott to Hotel Hilton and viceversa is $
30
. The odds that the conFerence will be held in Hotel
Mariott is
5 : 1
. The conFerence is a twoday event. So she needs to book a room For two nights. IF she stays
in the wrong hotel, she needs to take a taxi From the hotel she is staying to the conFerence venue and back
on day
1
and again book a taxi once on day
2
. Assume that the taxi charges to the airport From both hotels
are same, $
60
.
a)
(5 points)
Which hotel should she book a room in, iF she wants to minimize her risk or expected cost?
Explain your solution in detail.
b)
(10 points)
IF she decides which hotel she wants to reserve by ±ipping a Fair coin, with
Pr
{
heads
}
=
Pr
{
tails
}
=
1
2
, then show the risk set and the admissible rules.
4)
(5 points)
Consider a random variable,
?
?
U
(0
,
1)
. It is desired to guess the value oF
?
by making three
observations oF a random variable,
X
1
,
X
2
and
X
3
, where
X
i
?
Bernoulli
(
?
)
,
i
= 1
,
2
,
3
. Is
S
=
X
1
+2
X
2
+3
X
3
6
a suFfcient statistic For
?
? JustiFy your answer (
Hint:
See iF you can come up with two or more diFFerent
combinations oF
X
1
,
X
2
and
X
3
, that yield the same value oF
S
).
5)
(5 points)
A communication system transmits
0
’s or
1
’s with equal probability. Whenever a zero is transmitted
the channel corrupts the bit with noise which is a zero mean normal random variable with mean,
?
2
0
. IF a
1
is
transmitted, then the channel corrupts the bit with noise which is a zero mean normal random variable with
mean,
?
2
1
. The receiver makes
n
successive observations,
X
1
,
X
2
,
···
,
X
n
and makes a decision on whether
the transmitted bit was a
0
or a
1
. Show that
S
=
?
n
i
=1
X
2
i
is a suFfcient statistic For the receiver to make a
decision. You can use the Fact that iF
X
i
? N
(0
, ?
2
)
and
X
i
are iid, then the pdF oF
Z
n
=
?
n
i
=1
X
2
i
?
2
,
f
z
(
z, n
)
, is
given by
f
z
(
z, n
) =
1
2
n
2
?
(
n
2
)
z
n

2
2
e

z
2
, where
?(
n
)
is the Gamma Function defned as
?(
n
) =
i
?
x
=0
x
n

1
e

x
dx
(iF
n
is an integer,
?(
n
) = (
n

1)!
and
?
p
1
2
P
=
?
?
).
6)
(10 points)
Your company must make a sealed bid For a construction project. IF you succeed in winning
the contract (by having the lowest bid), then you plan to pay another frm $
100
,
000
to do the work. IF you
believe that the minimum bid (in thousands oF dollars) oF the other participating companies can be modeled
as the value oF a random variable that is uniFormly distributed on (70, 140), how much should you bid to
maximize your expected proft? Explain all the steps in your solution in detail (
Hint:
IF you want to win as
well as make proft you need to bid at least $
100
,
000
).
Answer