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Unformatted text preview: UCSD ECE109 Midterm Exam (051007) University of California, San Diego
Department of Electrical and Computer Engineering ECE109  Spring 2007
Midterm Exam  SOLUTIONS
Thursday, May 10, 2007 11:00am to 12:20pm
Location: CENTER 216
K. Zeger Name Your UCSD ID Number
Signature INSTRUCTIONS
This exam is open book and open notes. No calculators, laptop computers, or other electronic devices
are allowed.
Write your answers in the spaces provided. Show
all your work. If you need extra space, please use the
back of the previous page. Partial credit will be given
only for substantial progress on a problem. Zero
credit will be given for correct answers that lack adequate explanation of how they were obtained. There
is a maximum total of 30 points on this exam. Simplify your answers as much as possible and leave answers as fractions, not was decimal numbers. GRADING
1. 10 points
2. 10 points
3. 10 points
TOTAL (30 points) ...
Page 1 of 4 UCSD ECE109 Midterm Exam (051007) Problem 1 (10 points)
Al, Bo, and Cy tell the truth according to certain probabilities. If any two of the three of them tells
the truth, then so does the third one. The probability that all three of them tell the truth is and the
probability that all three lie is . If Bo tells the truth, then Al lies with probability . If tells the
truth, then the probability that both Bo and Cy lie is . What is the probability that Cy is truthful, but
neither Al nor Bo are truthful? (Leave your answer in terms of
.) ¡ ¢ ¤ ¥ ¨§¦¢
¥ ¤¥
¡ , £ , SOLUTION: Let
We need to ﬁnd denote the events that Al, Bo, and Cy are telling the truth, respectively.
and we are given:
(1)
(2)
(3)
(4)
(5) !£
©
©£ 8¤ #
)5$7£ 2 6©(
¢# 2
5$4© 3 £
#
$ © £
¡
# © $!£
0#
1)&"© £('$& £%$ "£
#
©# ©
, , !DC£(14F E"DC£(1BA !£([email protected]
© #
© #
© # . Then we have A B
0 x y s
0 0
z
C from (4)
¢
'# A ¢
¤
STR G TR i S
# e WG A G gb
9
#F #
1¨! © C£(
p
qR G ¢
S
S ¡ SVR
¤ h S ¡ TR
#$ © C£(
#$ f © dc('#$&aX%`XY£1# WG F G A G 9
£b
©
e
¡ STR
STR
¤
9 U G
¤'#
9
A H G
Page 2 of 4 S
VR @QP#
#9 I
¤
¢ S
TR BQP#
#A I ¢ Let from (5)
from (3) Page 3 of 4 e 3 Gb 3
e
8RS
e
S
@TR b ¨
e R S 3 Gb © 3
3 G S 3 b ©¨ 3 S #© S b¨
¨ 'TR #e
#
© 3 S
TR #
'£ 5 #
#
#
#
. Then ¨
e 3 G S 3b 3 S
©
HI3 EF¨
3 G4TR © G £
S
)
11 b 0( ¨
9876#% 2e ¢ ¨©q¢ © S £
¢
¤
D&e [email protected] % G ¢ & £ %
9876# "
8976# "
¢ &¥e db ¡ %
¢
9876# " Thus, equating (6) and (7) gives ¨
#
'£ 5
83
S
© TR
¨£
3 '#
© 4G
)(
0£¨ £
11
#% 2e ¢ ©¨b ¢ © S '#
¢ ¥e ¢ ©b¦¦§¥¢'© %
¢& £ % #
& ¨¤
#"G #"
¢
¥&e d¢b ¡ % # R
#
$"
2R (1#
Let (7)
(6) SOLUTION: We have ©
¨b ¦¤ ¢ £ ¢
e ¢ ©¦§¥£© G %# e db ¡ £ !R
#)
R ( 0
'# e ¢ b ¡ R ¥ 60 ¢ Let
for be a continuous random variable whose probability density function is
and
otherwise. Suppose
. Find and . Problem 2 (10 points)
UCSD ECE109 Midterm Exam (051007) UCSD ECE109 Midterm Exam (051007) Problem 3 (10 points)
Let
be a random variable whose probability density function
Find (and precisely describe) the cummulative distribution function is shown in the ﬁgure below.
of . ¡ e ¢ b e ¢b 1/2 . If 8 8
¢ S¢G 3 3 R 3 # e d( ¢b R
ES #¨
#EF &¥e 3
¦
R b §"
#
4F pF S R h 0
1# e ¢ b 3 ¥5©0
¡¢¡ ¡
¢¢ # e d( ¢b ¦" &3%
Page 4 of 4 ¢
RG R , then if 3 ¤¢
£ G R 3 # e d( ¢b . If and R
ES 3 R
# e d 0b Thus, if ¡ SOLUTION: Clearly 2 0¡
5¥¢ −1 , then ...
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This note was uploaded on 10/24/2011 for the course PSYC 109 taught by Professor Mookherjea during the Spring '11 term at UCSD.
 Spring '11
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