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Unformatted text preview: ECE 603  Probability and Random Processes, Fall 2002
Final Exam
December 17th, 10:30am12:30pm, Paige 202
Overview
The exam consists of six problems for 120 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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© §¥ 0( Some potentially useful information 1. [15] Two points are drawn independently and at random from the interval
. The outcome (or observation) for the experiment is the distance between the two points. Deﬁne a nontrivial probability space for this experiment; that is, ﬁnd
, where is the observation space, is a set
of subsets of to which probabilities are assigned, and is a probability
mapping from to
.
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e x x u W
x e x
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¤ Q ¡¥ e u e 2. Random variables
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[5] (a) Find c. is greater than
¤ b © , the probability that .
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[10] (c) Find
, the conditional probability density function for
given
(be sure to give limits!). in a ﬁeld experiment and ﬁnd that
b w ©Y ¥ ¤ 3. You are measuring a random variable
and
. b © W¥ [5] (d) Find , the marginal probability density function of
b [5] (b) Find
S bQ S bQ [8] (a) Suppose that the random variable
is input to your system, which
will malfunction if
. What can you say about
, the
probability that is greater than or equal to 12?
b ©
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u b [7] (b) Repeat part (a), but now assume you have one additional piece of
information: you know that is Gaussian.
b ©
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is generated as follows: I ﬂip a fair coin
repeatedly. If the ﬁrst ﬂips are “heads”, I let
; however, if
any one of the ﬂips before time is a “tail”,
. (Another way to
describe the same experiment: I ﬂip a fair coin repeatedly as long as I get
“heads” and record
for time ; however, as soon as I get the ﬁrst
“tail”, I then let
for all after that time). Does this sequence of
random variables converge? If so, in what ways and to what limiting random
variable do they converge?
! ©
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u ©X
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that is perfectly correlated over short intervals but then decorprocess
relates; in fact, he/she claims that
. Can such a process exist?
(Either give an example of such a process or show that such a process cannot
exist).
©X
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be a zeromean widesense stationary random process with
, and let
.
autocorrelation function
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. Note
that this gets complicated, but carry it as far as possible. Some of you
will be able to carry it the whole way; if not, indicate how you would
proceed from where you stopped.
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answer. widesense stationary? Be sure to fully justify your ©X
q¥ [12] (b) Let
be a zeromean widesense stationary random process with
autocorrelation function
, and let
.
¨ ©¨
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¤%
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)q¥ widesense stationary? If so, ﬁnd its power spectral density
. If not, ﬁnd the autocorrelation function
¤
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qR© ©X
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E¥© Estimate the power in
. (Note: Estimate does not mean guess.
There are multiple correct answers, but you must fully justify your answer.) ¤ 6. Note that you do not have to know anything about estimation to solve
this problem!
[15] The maximumlikelihood estimator (slightly modiﬁed to ﬁt this problem) is given as follows:
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argmax
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We study a Poisson (point) process for which the arrival rate is unknown.
Let be the number of arrivals during 5 seconds of observing the process.
Find the maximumlikelihood (ML) estimate of given that
. [Be
sure to fully justify your answer]. SX
T© q¥ m Is . u ¤ ¤ ...
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 Fall '10
 DennisGoeckel

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