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Unformatted text preview: EE 497B EXAM IV 5 May 2008
Last Name (Print): 5 olu‘llIJ OﬂS First Name (Print): ID number (Last 4 digits): DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO ,. Problem TWeight Score
r 1 25
2 25 
3 25 l
4 25  l
Total 100 l J
INSTRUCTIONS 1. You have 2 hours to complete this exam. 2. This is a closed book and notes exam. You may use one 8.5” X 11” note sheet and problem sets and their
solutions. 3. Calculators are allowed. 4. Solve each part of the problem in the space following the question. If you need more space, continue your solution
on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO
credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a
grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your aHSWers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must
show your work. Problem 1: (25 Points) 1. (13 points) Random variables X and Y have the joint CDF _ (1—ex)(1—e*y) $204420
FX,Y($7?J) — { 0 otherwise. (a) (4 points) What are the marginal CDFS, FX(:E) and Fy 7
Fxcq3= Fx’v(ql¢ﬁ:: (l—e 120 O o'ﬂérwuo—
 "7' #20
l 2.
.. 03 5
FY .. )J) i 0 otherWlbe‘ (b) (6 points) What are the marginal PDFs, fx(:r) and fy l3 (70: 31W“) : 57 no
x 17‘ 0 otherde 37‘ 63!. 20 , 51W) : [ 7; O atﬁemsﬂ (c) (3 points) Are the random variables X and Y independent? Justify your answer in a short sentence. gxnl (795.): W = 9:1 gﬁy‘ = Because 41%,)! (79;): ‘F'x (7‘) ‘CY (fl, {he (*0an don/lag”
X “Na Y we inﬂammam‘é. 2. (12 points) A different set of random variables X and Y have the joint PMF clw+yl w = —2,0,2;
PX,Y($)y): y:¥170713
0 otherwise. a 4 points What is the value of the constant c? PMF/4a? g 6; 69m.» = “' 60 4" ZC' 4‘ 6° =
1‘ T 'P
x’—z xw XsQL , J. : We :2? " w (b) (4 points) What is P[X < 1]? FfXLI] I PX” (.4) ’1) + pm, ('Zp) + pk,y(’21 1) + (hm/(0,1) + p” (0’ H)
_ s g L y,
’ '9“ 17 7 (c) (4 points) What is P[Y < X]? “:74sz 9x,y(f‘27"l) + (D)‘,\/(i'Z,03 + px;‘/("'ZJ I) 1' §~(0)_l) _ 7
+ 304'6 70 Fl. :— =c+Zc Problem 2: (25 Points) Random variables X and Y have the joint PDF /_ 1/2 431132131,
fX,Y(xv y) _ { 0 otherwise. 1. (5 points) Sketch the region of nonzero probability in the x,y—plane. 2. (6 points) What is P[X > O]? P1300] : L1 5:}; copay?” — L 1 .._.
[LX 7‘0 ‘1 3. (8 points) Determine “For «)1 4. (6 points) Find EU]: f —a3 1» l {:1 Al
2
o otkoruwbe, Problem 3: (25 Points) 1. (15 points) Random variables X and Y have the joint PDF ~(z+y) > 0 > 0
_ e 1' _ ,y _ 7
fX,Y($:1/) — { 0 otherwise. 0 (10 points) Determine the GDP of W : Y/X. FW ; L cu] '5 Y 5" w X] \/ y,wx
[[Ljétux
M x Prréwx] = g” [G’XMWQMOQy
z i on 591460; = j? [—357 «ix 0 0
w
4
6 O __w ‘ (w+37(
; Emame W1 = ‘ )on
d O  i Jam)?”
‘ 1‘“ + ﬁne =I  all?»
(7
o wéo
Flu/CW»: {l— u‘g, 0:20
o (5 points) Find the PDF of W = Y/X.
o (1240 “Cu/(w): (QR/(w) : I
&w (IA/+02" WZO 2. (10 points) Show that = I“: Y 15 (on‘EmeéJ (D E[ EfgCXMY] J 5 [ Eng‘HY] 4.1—» L { wcwa‘lﬂQx] £9360;
: [iftﬁ i 5‘60 LXIYC'Xlﬂ) Qx 1:..» and
3 co 00 1% (1,1) j‘xfiﬁ) $5”. 00%.] £1
W/f—J
5 Fx (7:) : jvgmﬂmex
; EEﬁOO] If; Y 35 ancKQJEPJ repute, MAE/rank out}! Summa‘éIc/JS. Problem 4: (25 Points) 1. (18 points) Suppose that X is a continuous random variable with variance 0%, and mean ,uX, and let Y = aX+b,
where a yé 0. (a) (10 points) Express Cov[X7 Y] in terms of the parameters a and 03". Cw Ext v1 = E[(x am (ya/n] Y: opxl—b
ED] = aELxyb = MACH: <10le Y] = E[(x—4x)(aX M)? j“ = OLE}:le ~ow/x E ngiwu/xE‘ X7 +0‘MX ‘ on {EEXZJ «w'ﬂ
:. 4L VarEXj Cal [797] ': 0“ V): (b) (8 points) Determine the correlation coefficient pxyy when a < 0 and a > 0. L Var EY] = Vow fat/X‘Fb] = «2' V¢YEXJ =¢1Tx ,_———————'""‘ s
V _ r!“ we
varrxzvaytyj V;— 43, Y—x'z. n1 )OX)’ 2: COVEXIYJ ORV—X2. LL22:
1 2. (7 points) Moment generating functions are useful for determining the moments of a random variable X. Let
X be a random variable with MGF ¢X(S) = E[6SX]
Show that dnqﬁ ( )
71 ~ X S ] ‘ dsn 5:0 ' ¢xm= Efesq= [ 6’JSXWCXG‘VQ7C Waco 7. §%es" PX (1)ch [I 30° x0 85" 9x ('70on ...
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This note was uploaded on 07/23/2008 for the course EE 497B taught by Professor Schiano during the Spring '08 term at Penn State.
 Spring '08
 SCHIANO

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