Systems 302
Tony E. Smith
SOLUTIONS TO PROBLEM SET 3
1.
(a) Let
i
x
= number of passengers in vehicle,
.
500
,...,
2
,
1
=
i
The
BLU estimator
for
the mean number of occupants,
)
(
X
E
=
µ
, is then given by:
500
1
1
1.778
500
i
i
x
x
=
=
=
å
(b) Construct the Bernoulli random variable
y =
î
í
ì
1
,
1
0
,
x
otherwise
>
Then
BLU estimator
for
)
(
y
E
P
=
is given by
500
1
1
.36
500
i
i
y
y
=
=
=
å
Partial Printout of JMPIN spreadsheet:
Rows
X
X_bar
Y
Y_bar
1
1
1.778
0
0.36
2
2
1.778
1
0.36
3
1
1.778
0
0.36
4
5
1.778
1
0.36
5
2
1.778
1
0.36
6
1
1.778
0
0.36
7
3
1.778
1
0.36
(c) If
j
X
= number of occupants in vehicle
j
= 1,2,°,12000 and
12000
1
,
j
j
T
X
=
=
å
Then
(
)
(
)
12000
(
),
j
j
E T
E X
E X
=
=
⋅
å
where
X
is occupancy of a randomly
sampled vehicle. Hence
12000
T
X
=
⋅
is a linear unbiased estimator of
(
)
E T
.
Moreover, since
±
var(
)
var( )
X
<
µ
for any other linear unbiased estimator,
±
µ
, of
(
)
E X
implies that
±
var(12000
)
var(12000
)
X
⋅
<
⋅µ
, it follows by definition that
that
T
is the unique
BLU estimate
of
(
)
E T
. In the present case, the desired
estimate is
12000
21,366
t
x
=
⋅
=
vehicles.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2.
For any estimator,
1
1
2
2
L
a X
a X
=
+
with
1
2
1
=
+
a
a
,
1
1
2
2
1
2
( )
(
)
(
)
(
)
E L
a E X
a E X
a
a
=
+
=
+
µ = µ
So all three estimators are
unbiased
. This means that we need only compare their
variances to determine which is
most efficient
. But by the independence of the
random variables
1
X
and
2
X
:
2
2
1
1
2
2
1
1
2
2
var( )
var(
)
var(
)
L
a X
a X
L
a
X
a
X
=
+
Þ
=
+
2
2
2
2
1
2
(
/35)
(
/105)
a
a
=
σ
+
σ
So we must have:
(i)
2
1
var(
)
105
L
σ
=
(ii)
2
2
2
1
1
9
1
var(
)
(
)
16 35
16 105
140
L
σ
= σ
⋅
+
⋅
=
(iii)
2
2
3
1
1
1
1
var(
)
(
)
4 35
4 105
105
L
σ
= σ
⋅
+
⋅
=
Þ
2
L
is
best
(most
efficient
)
3.
To compute the mean of
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 ese302
 Regression Analysis, BLU, WI, Mean squared error, Linear Unbiased Estimator

Click to edit the document details