practicefinalsols.pdf - Solutions 180B PRACTICE PROBLEMS...

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Solutions 180B PRACTICE PROBLEMS FOR FINAL Please simplify your answers to the extent reasonable without a calculator, show your work, and explain your answers, concisely. If you set up an integral or a sum that you cannot evaluate, leave it as it is; and if the result is needed for the next part, say how you would use the result if you had it. 1. Suppose Bob is trying to guess a specific natural number x * ∈ { 1 , . . ., N } . On his first guess he chooses a number X 0 uniformly at random. For t N , if X t = x * the game ends; if X t negationslash = x * , he guesses X t +1 uniformly at random from among the numbers different from X t . a. [5 points] What is the expected number of guesses it takes Bob to find x * ?
b. [5 points] Suppose Bob has a bad memory and can’t remember the number he guessed previously, so that he guesses X t +1 uniformly at random from among { 1 , . . ., N } . In this case what is the expected number of guesses it takes him to find x * ?
Solutions 180B PRACTICE PROBLEMS FOR FINAL c. [5 points] Suppose Bob has a good memory, and at each step guesses uniformly at random among the numbers he has not guessed at any previous step. Now what is the expected number of guesses it takes him to find x * ?
2. [15 points] 2 n N is a prime number if its only divisors are 1 and itself. The Prime Number Theorem says that the primes are distributed approximately as if they came from an inhomogeneous Poisson process P ( x ) with intensity λ ( x ) = 1 / ln x . [We have to say approximately since (1) the primes are integers, not general real numbers, and (2) the primes take determined, not random, values.] Use this theorem to estimate the number of primes in the interval [2 , N ].

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