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midterm 2 - (2 Let X n be the random variable that equals...

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Math 55, Second Midterm Exam Thursday, April 9, 8:10am–9:30am This exam is closed book. You may not use any books, notes or electronic devices. Please write your answers in a blue note book. Write your name, the name of your TA and your section time on the cover. There are five problems, each worth 20 points, for a total of 100 points. Answers without justification will not receive credit. Your graded exam will be returned in your discussion section on Monday, April 13. (1) Consider all permutations of the letters ABCDEFG . (a) How many of these permutations contain the strings ABC and DE (each as a consecutive substring)? (b) In how many permutations does A precede B (not necessarily immediately)?
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Unformatted text preview: (2) Let X n be the random variable that equals the number of tails minus the number of heads when n fair coins are Fipped. ±ind the expected value and the variance of X n . (3) Prove that log 2 (5) = 2 . 3219 ... is not a rational number. (4) Solve the recurrence relation a n = 6 a n-1-9 a n-2 with the initial conditions a = 1 and a 1 = 9. (5) A thumb tack is tossed until it ²rst lands with its point down, at which time no more tosses are made. On each tack toss, the probability of landing point down in 1 / 3. (a) ±ind the probability that exactly ²ve tosses are made. (b) What is the expected number of tosses?...
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