Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2010
Final Exam
December 14th, 10:30am12:30pm, Marston 132
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. ©
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a [5] (c) Find 2. Consider the following experiment: I ﬂip a fair coin. If the coin comes up “heads”, I record the
outcome (zero) in my notebook; if the coin comes up “tails”, I generate a random number from a
uniform distribution on the interval
, and record the result as the outcome in my notebook. As
has been the standard from class, denote the outcome of the experiment .
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, ﬁnd the probability that the coin came up “heads”.
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for partial credit.)
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. Determine whether the sequence
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probability, and convergence in distribution. You can do them in any order that you want, and you can
use one type of convergence to imply another when appropriate.
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they are correlated such that
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and then sum together the ﬁrst
elements of the sequence
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the variance
of , and . (Hint: Write the joint probability density function of and
as
, and use the deﬁnition of the expectation of
to get started. You can get to the
answer in just a few steps if you head the right direction.)
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convergence in distribution. You can do them in any order that you want, and you can use one type of
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converges, and, if so, to what and in
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2 [8] (b) Suppose
is input to a linear timeinvariant ﬁlter with impulse response
. Find the power at the output of the ﬁlter.
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is input to a linear timeinvariant ﬁlter with impulse response
. Find the power at the output of the ﬁlter. © s 5 i¥ 30 £
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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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