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Midterm04 - University of California San Diego Department...

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Unformatted text preview: University of California, San Diego Department of Electrical and Computer Engineering ECE109 - Spring 2004 Midterm Exam - SOLUTIONS Wednesday, May 5, 2004 6:30pm- 7:50pm Location: HSS 2250 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 40 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 4. 10 points TOTAL (40 points) ... 1 Problem 1 (10 points) £¡ ¤¢ Suppose the probability of tossing a Head for gold and silver coins is and , respectively. A box contains 2 gold coins and 3 silver coins. One coin is randomly taken from the box and that coin is tossed times. We learn that the coin came up Heads on exactly one of the tosses, but we don’t know which toss. Given this observation, what is the probability that the coin chosen was silver? Give your answer in terms of and . ¥ ¦ ¦ ¦   ¨ § © ¦ § G T f` as as as , i  r¥ i h¦ g i h¦ g g h¦ i y £¡ ¤¢ p q¥ g ECA x¤¥ D B  g3EC5x¤¥ D BA g D BA 3EC5x¤¥ [email protected] [email protected] 9 @ £ £  £   £¤¢ ¡ £¤¢ ¡ £¡ ¤¢  r¥ sq¥ p q¥ H P¡  i  f¤¥ 9 I p w¤¥ 9 I s w¤¥ 9 I g h¦ £  £   ‚ „ #ƒ y G 2 £¡ P¢ and therefore and therefore and therefore , s t¥   V XV ) ¢YWU PG for the 3 specific cases: £¡ P¢ ` aT   ¨ § ©    ¨ § © £ , so , so , so ebb !dcU T  £Pv ¡ £Pv ¡ £¡ Pv  €¥ pu¥ s u¥ 9I 9I 9 I then then then be the   ASB RQ ¦) G DB   ECA ¤¥ ¥ 87¦ H ¤¡P£ £ H P¡ 9 I  GFEC6¤¥ D B A ¥ 87¦ H ¤¡ 9 @ 63¨©5423#"!©10$%$#"!© 2  ¨ § )  ¨  ©  ¨ §   ¨ © ¨ § $%$('&©  § &©  ¨  ©  ¨ §  $%$#"!© What is your answer in the limit as £¤¢ ¡ £¤¢ ¡ £¡ ¤¢  €¥ su¥ p u¥ If If If tosses of the coin is Heads, and let , which is ¨ ¥ SOLUTION: Let be the event that exactly one of event that the chosen coin was silver. Then we want . . . ? for  ¤ tr ¦ us` q)  Y©B ¦ f 9 Y©B ¦ ¨ Yf ©B ¦ 9 Yf©B ¦ 9 I Y¨ ©B ¦ i YWD U ¨ phX0AA ©B ¦ S gQ  9 9  d E D eFAA Yf aA ©B ¦ 9 Y aA ¤ c©B §G b ¨ ¦ YWD U `X0VA E DB FAA S  C  SQ TRP FAAC E DB %2 I H ) '¦ ¢5G¦ R) '© 0 2 (  £ £ C B %999 3 "[email protected] 2 5&7 4 &G 52  4 "1R) '© 0 26 3 2 0 3 20 y (  ¨ ¦ ©B §G H  0¥r£¡  ¤ ¢  % &y  £ s   $ #"! ©  ¥   y p¤ y be a random variable with probability density function for . Find the probability that is positive. y 3 SOLUTION:   ¤ f7¥3¡ Let , and Problem 2 (10 points) Problem 3 (10 points)  ¤ 0¥ Let be a random variable whose cummulative distribution function below. Find the expected value of . is shown in the figure F(u) 2/3 1/2 −1 is ¤ £ y ¤ ¤ ‚ G ¡ v ¢¡ £v £¡ Pv „ #ƒ   ¤ 07r¢ ¡ is C ¤ 0¥ ¢ ¡ ¤  3 y  PG  ¡ ¢  £ ) Q   ¢¡ £v 4 ¨¦ ©§¥  ) 9   %  £¡ Pv   so the expected value of €9 SOLUTION: The pmf of u 10 2  ¤ Problem 4 (10 points)  ¤ 07 Let be a random variable whose cummulative distribution function below. Find the expected value of . is shown in the figure F(u) (1/2)(1−u2) u 0 is £ £ else y £ £ ¤ 9 ¤ ¤ S   0¤ y ) F   P¢ £¡  F  9 @ P¢ £¡ ¤ „ #ƒ  ¤ ¡ 07 ¢¡ is ¤ ¡ 07 ¥ b y £ £ ¦ 9 £ ¤ ¤ £ ¤ ¡ 7¥ ¢¡  ¤ 9 ¤b ‚ „ƒ £¡ Pv y else  y is B ¤ b 0¥w¨¡ ¤  ¤ ¡  ¥ ¨ ¥ © P ) q¤ b  E  3 6 ¡ ¢  S ¤ )G 5 ¤ b ¤  £¡ Pv D P B 9  % §  E BP ¡ ¢€9 D EB    ¤ so the expected value of y The pdf of ‚ SOLUTION: The CDF of 1 ...
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