Midterm07

Midterm07 - UCSD ECE109 Midterm Exam (05-10-07) University...

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Unformatted text preview: UCSD ECE109 Midterm Exam (05-10-07) 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 (05-10-07) 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 (05-10-07) UCSD ECE109 Midterm Exam (05-10-07) 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.

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