�
�
Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.432
Stochastic
Processes,
Detection
and
Estimation
Problem
Set
4
Spring
2004
Issued:
Thursday,
February
26,
2004
Due:
Thursday,
March
4,
2004
Reading:
This
problem
set:
Sections
1.7,
3.2.5,
3.3.1,
3.3.2
Next:
Sections
3.3.3
 3.3.6,
3.4
Exam
#1
Reminder:
Our
first
quiz
will
take
place
Thursday,
March
11,
9
am
 11
am.
The
exam
will
cover
material
through
Lecture
8
(March
6),
as
well
as
the
associated
Problem
Sets
1
through
4.
You
are
allowed
to
bring
one
8
1
��
×
11
��
sheet
of
notes
(both
sides).
Note
that
there
2
will
be
no
lecture
on
March
11.
Problem
4.1
Suppose
w
,
z
are
scalar
random
variables,
and
that
1
/
2

z

<
1
p
z
(
z
)
=
.
0
otherwise
You
are
told
that
the
Bayes
leastsquares
estimate
of
w
given
an
observation
z
is
1
−
1
/
2
z
�
0
ˆ
w
BLS
=
−
sgn
z
=
,
2
1
/
2
otherwise
and
the
associated
meansquare
estimation
error
is
�
BLS
=
1
/
12.
However,
you
would
prefer
to
use
the
following
adhoc
estimator
ˆ
w
AH
=
−
z
.
w
AH
) =
E
[
w
−
ˆ
(a)
Is
it
possible
to
determine
b
( ˆ
w
AH
],
the
bias
of
your
new
esti
mator,
from
the
information
given?
If
your
answer
is
no,
brieﬂy
explain
your
reasoning.
If
your
answer
is
yes,
calculate
b
( ˆ
w
AH
).
(b)
Is
it
possible
to
determine
�
AH
=
E
[(
w
−
ˆ
w
AH
)
2
],
the
meansquare
estimation
error
obtained
using
this
estimator,
from
the
information
given?
If
your
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 Spring '04
 Prof.GregoryWornell
 Electrical Engineering, Variance, Probability theory, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, linear leastsquares estimate

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