Midterm03

Midterm03 - 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 2003 Midterm Exam - SOLUTIONS Wednesday, May 7, 2003 5:00pm- 6:20pm Location: Peterson 110 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 160 points on this exam. Simplify your answers as much as possible and leave answers as fractions, not was decimal numbers. 1 GRADING 1. 120 points 2. 40 points TOTAL (160 points) Problem 1 (120 points) In room #1 there are 3 men, 5 women, 7 boys, and 9 girls, and in room#2 there are 2 men, 4 women, 6 boys, and 8 girls. A random person leaves room #1 and enters room #2. Then a random person leaves room #2 and enters room #1. Assume that a person cannot be both a boy and a man and a person cannot be both a girl and a woman. ¢£ ¢ SOLUTION:Since and ¡ ¡ (20 points) (a) Determine the probability that are the same person. went into room #2 there is one extra person in room #2. The probability is . 2  ¢ ¤ ¤© ¤¦ ¤ ¨§¥£ ¢ ¨ ¡  ¨ ¡£ ¡ £ ¤  £¤ © ¢ ¢   ¨ ¤¢ ¢ £ 3 the event that is a boy. Then we want  " #¢  ¤ !¤ ¦¢  ¤  ¡ is female, and is not a boy. SOLUTION:Let be the event that . We have is female given that (20 points) (b) Determine the probability that   ¨ ¡  ¢ ¥£ ¨ ¥ £ ©§¡¦ ¤¢ ¢ £¢ ¡ ¤ ¤© ¤¦ ¦ ¤£ ¤£ ¡ the event that ¢¤¤ "!¤ !¤   ¤ " ¤ ¡ 4 ¥ ¥ ¦  ©  §" ¢ £ £ ¤ ¢ ¢ ¢¤¤ !¤ !¤ " ¢ £¢ ¤  ¤ © ¤ ¦ ¤ £   !  ¨ ¤ ¡ ¢  ¨ ¡ §¡¥ £¤¢ ¢ ¤  ¤ ¡ ¦ £¤¢ ¤  £ ¡ ¤ £¤¡ ¢  ¤¢ ¥ £ £  ¤¢ £ is an adult, and ¡ be the event that ¡ SOLUTION:Let . We have (20 points) (c) Determine the probability that is an adult. Then we want is an adult (i.e. not a boy or girl). 5 ¨ ¦ £¡  ¦  £ ¢ ¥£   ¥  ¦ £ £   ¤¢  ¨ ¢ ¡ ¤¢  ¨ §¡¥ ¤¢ £ £ " ¦£ £ ¥£ ¦ ¥ £  ¥ £ ¦   ¥ ¡ ¤¢ ¢  ¥ ¨ ¡ ¤¢  ¢ £ £ £ ¢© ¦ ¥ £  ¦ ¢ ¥ ¢ £ ¦¢ ¥ ¢£  ¢  ¤  ¢ £¢ ¤  ¤ © ¤ ¦ ¤ £ ¢ ¤  £  ¢  ¤ ¢©  ¦¤  ¡ ¦   ¥ ¦ ¥ ¡ £ ¡  ¦  ¥ ¡ ¡ ¤  ¥ £  ¥ ¨ £¢  ¨ £¤¢ ¥ ¡ ¤¢ ¤  ¤ ¦ ¤¤¢ ¥ ¡ ¤¢¢   ¥ ¡ ¤¢ ¥ £ £¢  £¢ £ £ ¦   ¦¥    ¡ ¤¢ ¢  ¨ ¡ ¤¢  ¢ £ £ ¦ ¢¢ ¦¥   ¦ £ ¥ ¢ £ ¦ £ ¥ £ ¢  ¢¢ ¢ ¦ ¤   ¢ ¤ !¤ ¤ " ¢ £¢ ¤  ¤ © ¤ ¦ ¤ £ ¢ ¤ "!¤ ¤ ¢ £¢ £  ¤  ¤ "!¤ ¡ ¦  ¨ ¡¤¢  ¨ ¥ ¡ ¤¢  ¤  £¤¢ ¡ ¥ ¡  £¤¡ ¢   ¡ ¤¢ £ £ £ ¢ £  ¢ ¤  ¤ © ¤ ¦ ¤ £  © ¥ ¡ ¤¢ £  ¢ ¤¦   ¢ £ £ © £     © £    ¦  ¡ ¡  ¦ § ¢ £ © £ ©  £ ¨ £  ¦ § ¦ § £ ¥ £ ¤ ¢ ¢   ¤ "¤ ¤  ¤ £¢ "  ¨ §¡¥ ¤¢ £ is a woman, the event that . We know: ¡ ¡ is female given that ¡ ¡ SOLUTION: Let be the event that is female, the event that and the event that is a girl. We want to find (20 points) (d) Determine the probability that is a woman, is not a woman. (20 points) (e) Determine the expected number of males that are in room #1 after but before enters room #1. leaves room #1 ¡ SOLUTION: Let be the number of males that are in room #1 after leaves room #1 but before enters room #1. There are only 2 possible values can be, depending on the gender of . We have ¡ £¢ ¥  £¢ ¥   ¢ £  ¢ ¤  £      £   £ ©¨     ¡ " ¢¤  £ ¢   ¤ ¤ !¤ " £  ¢¢  ¦  ¥¢  §¢ !¤  ¢¢ £" ¡£ ¡ ¡  " ¤ £ ¢  ¤ "!¤ !¤   ¥¢  ¦ ¢  ¥¢  ¤!¤  ¢¢ £ £" ¡£ ¡ ¡  ¤  " £ ¨ ¦  male £¢ £ ¤ £ ¤ £ ¨ 6  ¢ ¢ female ¢ ¤ ¤© ¢ § ¥ ¤© ¤¦ ¤¦ ¤ girl girl ¢  © ££ ¨ £ ¨ £ ¨ £ ¨   ££ ¨ ££ ¤ £ ©   £ ¨ £ ¤       girl girl girl girl £       girl girl girl £  ¢   ¢  ¢ ¤ " £ ¤ " £ £ "      ¢ ¢£   £ £ £ ¢ £ ¤ £ ¦  £   ¦ £ £ ¤  ¦ ¢ ©£   ¦ £¥  £ ¦ £ £ £  ¦  ¢ £ ©  ¦ £ ¢ £ ¦ ¤  ¦ ¢ ££ £ ¤  £  ¦ £¥ ¦ £  ¤ £  ¦ ¤   ££   ¦¦ ¥  ££ ¨ £¢ ¨ ¨ £¥ £ £   £ ©   ¨ £¦  ¢£¥  "¥  ¤ £ ¢ ¤ £ ¥¦¤ "¤ ¤   ¢ ¤  ¤ © ¤ ¦ ¤ £ ¦¤ !¤ !¤ ¥ " ¤  £ "  ¥  ¡ !¤ ¤ ¢¢ ¤£  ¢ ¥¤   ¡ ¡ ¢¦¡ ¢¢ £ ¢ ¤¡ ¡  ¢¡ ¢ ¡ £  ¥¢¢ ¢ £¡ ¡ ¢ ¥ ¡¢¢ ¤ £ £ ¡ ¢ £  ¢ £ ¦  ¢ ¤  ¤ © ¤ ¦ ¤ £ ¥¦¤ !¤ !¤ ¥ "   ¤    ¡ ¡ "!¤ !¤ ¥¢ ¢ ¤ ¥ ¢ ¤ ¢ ¡ ¡¡ ¢¢ £ ¡ ¥¢  £ ¦ ¢  " ¥   ¢ ¤  ¤ © ¤ ¦ ¤ £ ¥ ¤ "!¤ !¤  ¦ £   ¥¢ ¡     girl girl girl ¢ ¡ £¡ ¢¢ £ girl girl girl girl  ¥ girl  © ¥¤ ¢ ¦  ¤ £ ¡ ¢ £¡ ¡ girl £ 7 girl girl      £ ©¨ ¡     £   £¢   £  £ ¨ ¥¢ ¡ ¢¢  £ ¢ ¤   ¢¢ ¡ ¡ ¥¢  £ ¢   ¥¢ ¡ ¡ ¦ SOLUTION:Let be the number of girls in room #1 after enters room #1. Let be the fraction of people in room #1 that are girls after enters room #1. The number of girls in the 1st room can either decrease by one (if is a girl and is a not girl), increase by one (if is not a girl and is a girl), or stay the same (if and are either both girls, or neither are girls). The original number of girls in the 1st room is 9. Thus, ¡ ¡ ¡ ¡ ¡ (20 points) (f) Determine the expected fraction of people in room #1 that are girls after enters room #1. ¡ £  ©   ¥¥¥ ¨¦¤ ¥ and ¡ £  ¥ Q§¢ ¥ ¤¦ ¨¦¤ ¥ odd for . ' 8¨¥ ¤ %# DC" [email protected] H@¤ WV `¤ XWV ¢ ¨¥ Q¦ ¤ ¤  B¨¦Q¦  C¤ 5 ¥¤ 5   B¨¥ Q 5¤ ¦ B¤ 5 '   ©  ¢ £¡        £ ¢ £¡ §¥¤ ¨¦£     £ £  5B¨¥ ¤Q¦  B ¤ 5 G ' ! %# &$" T 77 5 31 ) '1 ¤ U48 6P2SR¨¥ Q¦ P8 649¦IH¨¥ ¤ 77 5 31 ¥ '1 7 531 ¥ 8 649¦' EE E E %# DC" E E ¨¥ ¤ ! '   !  ¦ E F 7 531 ) 8 6420' ' 7 1¥ A B ¢ 8 65439¦' % # £ @¡ 8 654320' ( &$"   7 1) ! £    £  ¥¢ ¦   £  ¢¢ £ ¢   £  ¢¢  £ ¡ ¤£ ¡¤ ¡ ¤    £   is odd. for   £ 8 . odd odd odd  £ be the closest integer to . Suppose  £  £  ¢¢ ¡    ¢ ¤    be a random variable with probability density function . Let . £ Now take is odd let SOLUTION:For each Then the desired probability is (20 points) (a) Find the probability that the closest integer to £   Let for Problem 2 (40 points) ¢  ©  ¨ ¢§  ¢ @¡ ¡   ¢ ¢§ for  ¨¦¤ ¥ ¢   ¨ §¢ ¢¡ ¢ ©   5 #¥ " ¥ £ ! §    £ ¡¡ £  §  A ¥¢ A ¡     ¨ ¢§ ¢ a  ¨©¢§ ¡  ¦  £  9 for else gives ¢ @¡ ¡ ¤ ¨ ©¢§ . Suppose £ and  © ¢  ¢ ¢  ¨ §¢ ¢ £ ¥ ¦ ¡    39¥ 7 ¢ CS9¥  51  §  ¢ 5B¨¥  §    ¨¥ ¤  ¥  ¢§ £  A A %    £ ¡ A A   £ ¡ ¡ ¡ ¥¢   £ ¡ ¢¢ ¤ ¤ , so suppose  Substituting in ¡ ¥¢  £   ¦  £ ¡ £ ¤   ¢ £¤ ¢   SOLUTION: Clearly if then . Then,  ¢ (20 points) (b) Determine the probability density function of and . ...
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This note was uploaded on 04/07/2008 for the course ECE 109 taught by Professor Kennethzeger during the Spring '08 term at UCSD.

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