This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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
n=&_& /‘\/‘\/‘\
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‘;
A lamemoaoro a We ‘0 a
i a t r ,twtm Wl“ o g “m w} k mm MW Amara,ij A“ " V‘r w f g
I? 3: “ “ is?) £1th
t
«w , , WWWW r s
“W [4%“riaa3i
,,,,, , 3 ram if“
fax kw.“ my: 1 n“ M 5 It: kwkj” erg g] r;
A ‘ W,“ ' 2 W Max NW ] gt 66:5;
Mr ’ ,  A [X f" ,3, g] {V} { f]
C g,,\g( \[ﬁ'] 2 \T: >134, [ “i” Lu, 7"“ if i “t” ( )r" ~« ‘
W 4 E xx": j 4 (: [Mm M] {n J} (“[M
kavtirrs§ei <7<ri a! #0 it)“/’ 3
“f “3 a {3
r m é §
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?“
x W “ Z36;
; w: j) x 23‘ k“
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 ...
View
Full Document
 Spring '08
 GELFAND
 Normal Distribution, Probability theory, probability density function, Cumulative distribution function

Click to edit the document details