4203hw6 - Combinatorics(MAD 4203 — Fall 2010 Due Monday...

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Unformatted text preview: Combinatorics (MAD 4203) — Fall 2010 Due Monday 11/15 Homework #6 Linearity of expectation is a very simple fact with a fancy name; it says that if X and Y are both random variables (i.e., their values depend on some random process), then the expected (average) value of X + Y is the expected value of X plus the expected value of Y . Note that this holds no matter what, even if X and Y depend on each other! This frequently makes computing expected values very easy. 1 We showed in class that the probability that a permutation has no fixed points (i.e., the probability that it is a derangement) is very close to 1 / e . What is the expected number of fixed points in a permutation of [ n ] which is chosen uniformly at random? Hint : What is the probability that π ( i ) = i in such a permutation? 2 A left-to-right maxima in a permutation is an entry that is greater than all en- tries to its left, i.e., π ( i ) is a left-to-right maxima if π ( i ) > π (1) ,π (2) ,... ,π ( i- 1). What is the expected number of left-to-right maxima in a permutation of1)....
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