Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 8
Topics: Covariance, Estimation, Limit Theorems
Due: April 26, 2006
1. Consider
n
independent tosses of a
k
sided fair die. Let
X
i
be the number of tosses that result
in
i
. Show that
X
1
and
X
2
are negatively correlated (i.e., a large number of ones suggests a
smaller number of twos).
2. Oscar’s dog has, yet again, run away from him. But, this time, Oscar will be using modern
technology to aid him in his search: Oscar uses his pocket GPS device to help him pinpoint
the distance between him and his dog,
X
miles.
The reported distance has a noise component, and since Oscar bought a cheap GPS device
the noise is quite significant.
The measurement that Oscar reads on his display is random
variable
Y
=
X
+
W
(in miles)
,
where
W
is independent of
X
and has the uniform distribution on [

1
,
1].
Having knowledge of the distribution of
X
lets Oscar do better than just use
Y
as his guess of
the distance to the dog. Oscar somehow knows that
X
is a random variable with the uniform
distribution on [5
,
10].
(a) Determine an estimator
g
(
Y
) of
X
that minimizes
E
[(
X

g
(
Y
))
2
] for all possible mea
surement values
Y
=
y
. Provide a plot of this optimal estimator as a function of
y
.
(b) Determine the
linear
least squares estimator of
X
based on
Y
.
Plot this estimator
and compare it with the estimator from part (a).
(For comparison, just plot the two
estimators on the same graph and make some comments.)
3.
(a) Given the information
E
[
X
] = 7 and var(
X
) = 9, use the Chebyshev inequality to find
a lower bound for
P
(4
≤
X
≤
10).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 MuntherDahleh
 Systems Analysis, Covariance, Probability, Standard Deviation, Variance, Probability theory, Oscar

Click to edit the document details