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Unformatted text preview: THE UNIVERSITY OF TEXAS AT AUSTIN
Dept. of Electrical and Computer Engineering Midterm #2
Date: November 11, 2010 Course: EE 351K Name: Last, First 0 The exam will last 75 minutes.
0 Closed book and notes. You may consult two double sided page of handwritten notes. 0 No calculators are allowed. 0 All work should be performed on the exam itself. The backs of the pages may be used
as scrap paper. To avoid confusion, please highlight your ﬁnal solution in the space
provided. 0 Fully justify your answers for more rational grading. 0 Good luck! Problem Point Value Your Score
1 10
2 10
3 10
Total 30 Problem 1. [10 points] Power generated by a wind turbine is a function of the speed of
wind Z 7 and can be expressed as X : 50¢? The speed of wind Z is a random variable with uniform distribution on [75, 125]. (a) (4 points) Find the PDF and CDF of X. 615?: View : "P (X s s)
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‘5!) ( 0m€a¢N~£$‘ﬁ (b) (2 points) The turbine is required to be capable of generating powers in excess of 500
(Watts). We conduct a single trial in which we measure the turbine power. What is the probability that the turbine meets the requirement in this trial? [In other words,
ﬁnd P(X > 500).] ’PLX '>500\:A‘ (llK (300') (C) (4 points) Now suppose that we are allowed to conduct n trials, If in any one of these
trials the turbine generates power in excess of 500: it passes the test. The speed of
Wind is independent from trial to trial. What is the smallest integer n such that the
probability of passing the test is higher than 0.9? {(0131 KPC‘l <1) vs S! 3 R 0’ 3% P (xi (\f )M “P c «r > 5%» A ~ a. (gem  l) (WX (Xxiw‘) X’Wl <7) ”(A LMUf‘ XLAjHMI AWN!) Problem 2. [10 points] Duration of a phone call is an exponential random variable with
expected value 3 min. All phone calls have independent duration. In a one—week period
someone makes 32 calls. The total calling time is a random variable T. (a) (3 points) What is the expected duration HT of all calls combined? What is the variance
0% of the duration of all calls combined? L.“ T’W‘ 6$§§C%’L,&kw%\ in)“ R‘V’ i ELALS : «1. I & \jcuvt' Lx‘; :. 7 u: I (b) (3 points) Use central limit theorem to approximate the probability that the duration
of all calls combined is more than two hours (i e. approximate P(T > 2 hours)) Write your answer in terms of the standard Gaussian CDF <I>(t )2 f2; [:00 6—7dt m
2:5“
,, (N , , a ——,~. p, . AN: \ ‘
PM > has, ‘; E3 t i > Air: M \) ‘9 (KM) EAL/W
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T133 ”m :2) f {Aha (c) (4 points) Now suppose that a person makes both voice and internet (modem) phone Fallc D'Ir‘nko Mkn‘no lo] 10 1V7n1ﬂo h‘ln QDQll 10 H R n cans. i lovability that aparticular“ plume can is M W p110 ie can is U. u. uration of a voice phone call is an exponential random variable with expected value 3 min.
Duration of an internet phone call is an exponential random variable with expected
value 9 min. Suppose that we are told that the length of a particular phone call was
X. For which values of X should we conclude that it was a voice phone call? Please
provide the answer which minimizes the probability of error. (330.6 25 ”1sz
A xi ’3) ’21 M 4W ate, 4 y __ 7L1 )x
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sz.gg Problem 3. [10 points] A weather satellite takes an image of the region of an ocean where
a hurricane is developing. We are interested in locating the eye of the hurricane (i.e., its
central region). Any location in the image has a probability p of being in the eye of the
hurricane. We employ a detector with the following response: conditioned on the eye of the
hurricane being present at some position, the detector generates signal which is Gaussian
with mean 1 and variance 02; conditioned on the eye of the hurricane not being present7 the
detector generates signal which is Gaussian with mean 0 and variance 1. (a) (3 points) Let X denote the signal generated by the detector. Find the mean and
variance of X. ELK} 1 El)” brig PLﬁjil 4r E [X \Msél, @191 PUMA (”JK6)
3’— ? '4’“ h“?) ' O : F Etﬂml 2 l3 EiﬁLlﬁel+ [UTA EEXll«mlc exit}
= 1) than + on?) . I” , 7— , ' .
irM‘CWl ‘> E1303 v MattiH 2 ”purse?“ +U~ﬂ ° F1: 31 ﬁbril—f” (b) (2 points) Find the conditional probability that a position in the image belongs to the
eye of the hurricane given that the response of the detector X > 10. Write your answer in terms of p, a, and the standard Gaussian CDF @(t) = %; ffoo e‘édt. mime) .WWWW‘ W. lD’CXNwEl VLF—l + ‘P UbiolE SHE) (c) (5 points) Suppose that we observe Y = X + W, Where W is a Gaussian noise N(O, 1), independent of X. Compute the linear least—squares estimate (LLSE) of X given Y,
and the corresponding mean—squared error. A , wWﬁY}  a.
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XL E: l VMQY) L l EEX1:?
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COX/L)“ X) ; {L‘OVCX‘ X+ V733 : VM (X) I A+’?€2_?Z VMUW 2 thMme > Q‘s—P31“? ...
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