15b_00f_me

# 15b_00f_me - Be sure to explain how you got your answer...

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Math 15B (Bender) Midterm Exam 27 October 2000 Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but calculators ARE allowed. You must show your work to receive credit. 1. (36 pts) A ﬁve digit number is a sequence of ﬁve digits, the ﬁrst of which is NOT zero. Thus, 12345 and 10101 are valid but 01234 and 1234 are NOT valid. (a) How many ﬁve digit numbers are there? (b) How many ﬁve digit numbers have all digits diﬀerent? (as in 12345 but not 10101) (c) How many ﬁve digit numbers have no digit appearing just once? (So 11111 and 10010 are okay, but 10111 and 12312 are not.) 2. (16 pts.) Nine people, including Alice, are to be divided into two teams of four people each, plus a referee. If all divisions are equally likely, what is the probability that Alice is the referee? (No, it doesn’t matter if the teams are distinguishable or not.)
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Unformatted text preview: Be sure to explain how you got your answer. 3. (24 pts.) An integer k from 1 to 9 is picked uniformly at random. Let X ( k ) = 1 if k is odd and X ( k ) = 0 if k is even. Let Y ( k ) be the remainder when k is divided by 3. (a) Draw a table like the one here and ﬁll in the probabilities. X \ Y 0 1 2 ? ? ? 1 ? ? ? (b) Compute Cov( X, Y ). 4. (24 pts.) Suppose a strictly decreasing function f : { 1 , 2 } → { 1 , 2 , . . . , n } is chosen uniformly at random. The random variable X is deﬁned by X ( f ) = f (1). (a) Describe choosing f in terms of choosing subsets of a set. (Specify the set, what subsets are chosen, and how they are chosen.) If S is a subset associated with f , what is X in terms of S ? (b) Derive the formula P ( X = k ) = ‰ ( k-1) – ( n 2 ) for 1 ≤ k ≤ n , for k < 1 and k > n . END OF EXAM...
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