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Unformatted text preview: COMER
April 19, 2010 ECE 302 Exam 3 1. Enter your name and signature in the space provided below. Your signature certiﬁes that you will not
engage in any act of academic dishonesty during the taking of this exam. 2. You may not use a calculator or any other reference materials. 3. This exam consists of three types of problems. For the multiple choice problems, no partial credit
will be given. Each ﬁllin—thebank answer will be scored either 0, 3, or 5 points. For the ﬁnal two
problems, partial credit will be given at the discretion of the instructor. Name: SOLUTio/ """" Signature: No Partial Credit
You must CLEARLY select one answer for each problem. 1. (5 points) Which of the following statements about continuous random variables X and Y, with joint
probability density function f Xy(111', y), is NOT true in general? (a) If X and Y are independent, then X and Y are uncorrelated.
@‘Elf X and Y are uncorrelated, then X and Y are independent.
(c) If X and Y are independent, then ny($, y) = f X (:c) fy(y).
(d) If X and Y are jointly Gaussian and uncorrelated, then X and Y are independent. 2. (5 points) Which of the following properties does NOT necessarily hold if X and Y are uncorrelated?
(a) The covariance Cov(x, y) = 0 (b) E[XY] = E[X]E[Y]
(c) The correlation coefﬁcient p Xy = 0 m, z
6;)”, E[XY] 0 3. (5 points) Which form of Bayes’ formula would you use if X is continuous and Y is discrete? (Note
that)0 represents a marginal or conditional probability density function, depending on its subscript). 2‘ P Y:
(a java» = M = x) = “I‘m/(j,ng 1") (mhuwm=&%%§@
c>nmwm=ﬂﬁ%%%&@
(d)p(Y=yX:$)=Mijg(/X_=gg<z=_w 4. (5 points) Given continuous random variables X and Y with joint probability density function f Xy (as, y) and joint cumulative distribution function Pkg» (9:, y), which of the following is NOT necessarily equal
to the probability P(a < X S b). (a>,thE:;.ﬁXY<x,y)dydw
(b) limy—ioo [FXi/(b, y)  FXi/(a, y)i
@ﬁamm @i>FXY(baZ/) ‘FXY(aay) Cit/5: \f/rxj: (MD
0 1y distributed between 5. (5 points) Let X and Y be independent discrete random variables each u
0 and 3 (i.e., the marginal probability mass functions p X (ac) and py(y) are constant for m = 0, 1,2,3
and y = 0, 1, 2, 3, respectively). What value does joint probability mass function p Xy(ZE, y) take for a: = 0,1,2,3;y = 0, 1, 2,3? WM) I j.
(a) 4 i/ '''' “ ?C "I “:2: r r
(p) 16 i x) (X: 0/} «Di ‘ ﬂ No Partial Credit You must CLEARLY select one answer for each problem. 6. (5 points) Three of the following quantities are equal to each other for any random variable X. Which
quantity is NOT equal to the other three in general ? (Note that m X = E[X]). (a) Var(X) (b) E[(X—mX)2] f F v (((((( I” ) . ,2” W
@ymegy » r. 2‘? o Final answers will be assigned 0, 3, or 5 points You must put your ﬁnal answers in the boxes provided. 7. (5 points) The joint probability mass function of random variables X, Y, Z is given by pXYZU‘» 27 : pXYZ(27 17 = pXYZ(2a 2: : pXYZ(2> 3) 2 Find ElXYZ]. i I w M J g , 4M 5 2 i Answer: 8. (5 points) Suppose that X, Y, and Z are independent random variables that are each equally likely to
be either 1 or 2. Find theiprobability mass function of the random variable X 2 + YZ. ‘ A ’L \ A i A I \ I . v I w
ﬁg "r Z)” 9’. g _ ‘ X9??? (xiieriZ
ttrreaartvanmweaewepaa. H, 2
Answer: VT) i 2“. 2: ES“ 2:“ 9. (10 points) Let X and Y be continuous random variables with joint density function l i J
’21 i 1 g; £+c, if0<x<1,1< <5 ,«t w fXY($,y) = 5 y . y 4’32” i O, otherw15e , , a
(2/, ZVZW, where c is a constant. (a) (5 points) Find the value of c. i. a) . .i 3 5;} [%+{i> ii“? ar+ifi Z (I? Final answers will be assigned 0, 3, or 5 points
You must put your ﬁnal answer in the box provided. L— a. (
(b) (5 points) Find P(X + Y > 3 . > a 1:7 . ‘ I ,, ‘l a; W E; ” A e" é“ x
l W. m WWW‘W; WWAWW X:
/i 59 t K Answer: 3 i [:9 10. (5 points) Let X and Y be continuous random variables with joint cumulative distribution function FXy (x, y).ﬁFind an expression for P(a < _<_ Z), Y 5,0!) in terms; of FXY (nan ). M) a I ,,
{414 t9 U ,) . W a w an iii/2960 K b
mm MR,“ W (Q) l 1. (5 points) Let X and Y be discrete random variables with joint probability mass function p Xy (— 1, — 1) = PXY(—1,1) = pXY(1,1) = IBM/(1,1) = 1/6; PXY(0a“1) = PXY(_170) = I’m/(0,1) =
gun/(1,0) = 0; and pr(0, 0) = 133. Find the marginal probability mass function of X. X f: ‘3’ WK“ I!) i) at L Y? Answer: Did/0) "LV®[>:: w i u: W2“ alplé“g; Partial Credit Problem You must completely justify your solution to get ﬁill credit. Partial credit will be given at the
discretion of the instructor. 12. (20 points) Let X and Y be independent continuous random variables with Y being exponential with
parameter 1 (Le, fy(y) = e“yu(y)) and X being uniform on [1,2] (i.e., fX(:c) is constant on [1,2]
and 0 elsewhere). Find P(ln(1%—) > 1). You may need the relation f jail—$2111; : tan—1 m + C for
some constant C'. L}, Partial Credit Problem You must completely justify your solution to get full credit. Partial credit will be given at the
discretion of the instructor. 13. (20 points) Let X and Y be continuous random variables, jointly uniform on {(33, y) : O < a: 3 2,0 <
y S 2} Ge, the joint probability density function of X and Y is constant for 0 < x g 2, 0 < y S 2,
and 0 elsewhere). Find the cdf and pdf of Z = %C%%, where min(ac,y) and max(:r, y) are the minimum and maximum of :13, y, respectively. ...
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 Spring '08
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