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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2008
Final Exam
December 15th, 10:30am12:30, LGRT 0321
Overview
The exam consists of ﬁve problems for 110 points. The points for each part of each problem are given
in brackets  you should spend your two hours accordingly. The exam is closed book, but you are allowed three pagesides of notes. Calculators are not allowed.
I will provide all necessary blank paper. Testmanship
Full credit will be given only to fully justiﬁed answers. Giving the steps along the way to the answer will not only earn full credit but also maximize the
partial credit should you stumble or get stuck. If you get stuck, attempt to neatly deﬁne your approach
to the problem and why you are stuck. If part of a problem depends on a previous part that you are unable to solve, explain the method for
doing the current part, and, if possible, give the answer in terms of the quantities of the previous part
that you are unable to obtain. Start each problem on a new page. Not only will this facilitate grading but also make it easier for you
to jump back and forth between problems. If you get to the end of the problem and realize that your answer must be wrong, be sure to write “this
must be wrong because . . . ” so that I will know you recognized such a fact.
Academic dishonesty will be dealt with harshly  the minimum penalty will be an “F” for the course. l
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a 2. If are jointly Gaussian random variables, each with zero mean, then: 1. A random variable is jointly Gaussian with itself.
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© ¢
£ ¡¥ 1. For the probability space
, let
and
(restricted to
, of course). The
subset
of
is deﬁned as the interval
, with the removal of the set of rational
numbers in
, and with the removal of the set of irrational numbers in
, where
.
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fv § © [10] (a) Starting from the basic deﬁnition that the Borel ﬁeld is the algebra generated from all
intervals
such that
, show that
is an element of the Borel ﬁeld.
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¨¥ [5] (b) Now, deﬁne . [5] (c) Now, deﬁne l
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¦ Find . 2. A single random variable:
T be uniformly distributed on the interval
. Find the probability density function
[5] (a) Let
of the random variable
. A very short sentence of justiﬁcation is sufﬁcient.
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otherwise
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) 2
3 , the median of be given by:
a a For a random variable
Find the median of . [7] (b) Let the density of the random variable . a 2
5 [8] (c) For an arbitrary random variable
with probability density function
, the minimum
mean squared estimate (MMSE) estimate of (without any other information such as ) is the value
the minimizes
. Show that
.
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a T© V 5 a¥R
p V 3. Let be a random variable representing the number of bad chips in a shipment shipped from Xcompany, and let be a random variable representing the number of bad chips in a shipment shipped from
Ycompany. The probability assignment functions of these two random variables are given below:
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The probability that a shipment (or set of shipments) comes from from Xcompany is 0.25 and the
probability a shipment (or set of shipments) comes from Ycompany is 0.75. Given the company,
different shipments contain an independent number of bad chips. Note: Some of the fractions get a
little messy  sorry!
[6] (a) Assume you receive a single shipment. Find the probability that the number of bad chips in the
shipment is less than or equal to 3.
[6] (b) Given the number of bad chips in the shipment is less than or equal to 3, ﬁnd the probability
that it came from Ycompany.
[6] (c) Suppose that you are going to receive a set of two different shipments from the same company
(Xcompany with probability 0.25 and Ycompany with probability 0.75). In the ﬁrst shipment, you
observe less than or equal to 3 bad chips in the shipment. What is the probability of less than or equal
to 3 bad chips in the second shipment?
[7] (d) Suppose that you are going to receive ten shipments from the same company (Xcompany with
probability 0.25 and Ycompany with probability 0.75). What is the probability that you observe less
than or equal to 3 bad chips in at least half of the shipments? (Just write the expression. No need to
evaluate.) eio2
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1i¥ 4. Let
be a widesense stationary Gaussian random process with mean
and autocorrelation function
, where
is the Dirac delta function. The random process
is
run through a ﬁlter combination as shown below, where the ﬁrst ﬁlter has frequency response:
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©W
ei¥
1 © . (Note: that the integral should look familiar, so simplify ©W
1i¥ ©
4E¥ 2 T©W
U1i¥ T©W
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1 © p
R ©W
1i¥
¨ a p R © ¦ ©W
1i¥ 2
W¥
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ei¥ . ¦ .
©¥
1 .
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©
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¢o2 ¡¥
¢o2 ©W
ei¥
1 ©W
ei¥ 2
4 5 ei¥
©W a © ©W
pei¥
© ©
¡¥
¢o2 ©W
ei¥ Find the probability density function for
work.) ¦
© W i¥
W
© ¥ widesense stationary? If so, ﬁnd its power spectral density in terms of 2 for ©W
1i¥ and autocorrelation function in terms of
©
4E¥ ( # ©W
ei¥ Find the mean function
. in terms of
©
4E¥ ©W a widesense stationary? If so, ﬁnd its power spectral density in terms of Find the probability density function for [20] (b) Let for
1 and autocorrelation function ©W
1i¥ a ¥ 1 . . Find the mean function
. Is . be a zeromean widesense stationary Gaussian random process with autocorrelation funcand power spectral density
. [10] (a) Let Is ¢ © 2 [10] (b) Find the power
in
your answer as much as possible.)
5. Let
tion a in ©W
1i¥ [5] (a) Find the power 4 and the second ﬁlter has frequency response otherwise . . (Hint: This is going to take you a little bit of ...
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This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).
 Fall '10
 DennisGoeckel

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