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Unformatted text preview: Final Exam of ECE302, Prof. Wang’s section
3:20e5220p1n Thursday, May 5, 20117 CL50 224. . Please make sure that it is your name printed on the exam booklet. Enter your
student ID number, e—mail address, and signature in the space provided on this
page, NOW! . This is a closed book exam. . You have two hours to complete it. The students are suggested not spending too
much time on a single question, and working on those that you know how to solve. . There are 16 pages in the exam booklet. Use the back of each page for rough work. . Neither calculators nor help sheets are allowed. ‘2 x“, at This aft; M
Name: fawnimeme
Student ID:
E—mail: Signature: Question I: [20%, Work—out question] Consider a random variable X. The probabilities
X = 1 and X : —1 are 1/2 and 1/2, respectively. We are not able to directly observe X.
Instead, we observe Y : X — N where N is a random variable with probability density
function fN(n) = 0.5647”. (1) X and N are independent. 1. [10%] Find out the linear MMSE estimator of X given Y : 3/. That is, your answer
should be a function thﬁmsﬂy). Note: If you do not know how to solve this problem, write down what MMSE stands
for and you will receive 4 points. 2. [10%] Find outAthe MMSE estimator of X given Y = y. That is, your answer should
be a function XMMSE(y). Note: If you do not know how to solve this problem, answer which estimator has
better performance: a MMSE estimator or a linear MMSE estimator. You will
receive 2 points. cm "ﬁfmﬂm < NM
g ‘2 Question 2: [10%7 VVorkout question] X1, X2, and X3 are three independent standard
Gaussian random variable. Construct three new random variables by Y1=X1+X2+X3
Y2=2X1—X2—X3
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iPOJM 1. [4%] Find the mean vector and the covariance matrix of (Y1, Y2, 2. [4%] Write down the joint pdf of (Y1, Y2, Note: If you do not know the answer to this question, write down what type of
random is the (marginal) random variable Y1. You will receive 2 point. 3. [2%] Write down the condition pdf Y2 given Y1 = 5 and Y3 = 2. a '= t ,F” _ [vu‘ ‘ , f‘;
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ﬁ 0 A d I » ww 3 ~ ,0 Question 3: [5%, Work—out question] 1. [4%] X is a Poisson random variable with parameter a = 3. Y is a Poisson random
variable with parameter a = 5. X and Y are independent. Let Z = X + Y. Write
down the probability mass function pk of the Z random variable. Also write down the characteristic function of Z. Note: If you do not know the answer to the above question, you can instead assume
that both X and Y are independent binomial random variables with parameter
n : 2,}? = 1/3. You can still get 3 point. 2. [1%] What does the acronym “i.i.d.” stand for? /z, (j... (I, m VA‘Aa a Question 4: [10%, Work—out question] Deniographically, 10% of the total population of
West Lafayette are Purdue students. To conduct an opinion poll, a statistician randomly
Chooses 2500 residents of West Lafayette. What is the (approximate) probability that
more than 270 (out of the total 2500 samples) are Purdue students? Note: You may need to use the facts that Q(1) = 0.15877 Q(4/3) 2 0.0912, 6203/3) =
0.0478, and = 0.0228. ‘ x
f " r ‘ New WT“ ages) p :::> a we P <” f “‘3 gig?“
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l < 5953M ““ “Warm
i’ Question 5: [15%, Work—out question] A random process X (t) can be described by X (t) =
cos(7r(t + 8)), Where 6) is a continuous random variable that is uniformly distributed on
the (0,1) interval. 1. [7%] What is the probability X (0.5) > 0?
2. [6%] What is the correlation between X (0.5) and X (1)? Note: You may need to use the following trigonometric formulas: cos(a ~ 5) — cos(oz + ﬂ) 8111(a) sin(ﬂ) : 2 (5)
sin(a) cos(ﬁ) = W (6)
008(0)) 008(3) 2 W (7) 3. [2%] What is the deﬁnition of “correlation coefﬁcient”? Question 6: [10%, Work—out question] X is a Bernoulli random variable with parameter
p : 0.3. Conditioning on X = :50, Y is uniformly randomly distributed on the interval
(0, l — 5130). Find the probability P(X + Y < 1). ., l. 33“) “AW Question 7: [10%, Work—out Ligation] [" l. [5%] X is random variable with” : 1/4. Find the conditional
1 expectation E(XX g 2. [5%] X is a binomial random variable with n 2 2 and p 2 1/3. Plot the correspond—
ing Cdf for the range of :c = —1 to 3. Please carefully mark the solid and empty end points of your piece—Wise curve. Question 8: [20%, Multiple choice question. There is no need to justify your answers] 1. [3%] X and Y are uniformly distributed in a unit square (0,1) x (0,1). We know that Z 2 X + Y and W = X — Y. Are Z and W independent? 2. [3%] X and Y are independent Gaussian random variables with mx 2 1, 0% = 1, my : —3, and a; = 1. We know that Z 2 X + Y and W 2 X _ Y. Are Z and W
E independent? : 3. [3%] X is a Gaussian random variable with mX : 0.1, 0% = 1. Is the following
[\j statement correct? “By the Markov inequality, we must have P(X 2 1) g =
l 0.1.” E 4. [2%] Is the following statement true? “A 99% conﬁdence interval is smaller than a
95% confidence interval.” variable X27 and X2 is independent of X3. Is the following statement true? “X1 f. [2%] Suppose we know that a random variable X1 is independent of another random and X 3 must be independent.” X2 and X3 are also uncorrelated. Is the following statement true? “X 1 and X3 must
be uncorrelated.” M 6. [2%] Suppose we know that two random variables X1 and X2 are uncorrelated; and j 7. [2%] Is the following statement true? “Since a Maximum Likelihood (ML) detector
\‘ maximizes the likelihood function, it outperforms the MAP detector.” 8. [3%] ls the following statement true? “A cumulative distribution function FX(.’17) is
1: always non—decreasing and right continuous.”
t ...
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 Spring '08
 GELFAND

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