zzzz-'s blog

Just for fun
By zzzz-, history, 6 years ago, In English

We all know what a permutation is. So, I remember a good problem from school. Let P be a permutation of 1, 2, 3, .... n. We define the function f(P) as the number of positions where P[i] = i (also known as fixed points I think). What is the expect value of f :)

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6 years ago, # |
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I think there is a O(n) solution using dp, let dp[i]: number of ways that i numbers can arrangue that there isn't a p[j]=j, then your answer is $$$\sum\limits_{i=2}^{n} (n-i)*dp[i]*comb(n, i)/n! + 1/n!$$$, and $$$dp[i]=(i-1)*(dp[i-1]+dp[i-2]), 2 \leq i$$$

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    6 years ago, # ^ |
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    As much as I remember there is also an O(1) solution, good dp :O

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6 years ago, # |
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Let p(i) be the probability that i is a fixed point, then p = (n-1)!/n!. Using linearity of expectation, f(P) = sum of p(i)*1 = |P| * 1/|P| = 1

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    6 years ago, # ^ |
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    Nice :)

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    6 years ago, # ^ |
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    I know this works but this feels so unnatural to me.

    It really feels like $$$E[X + Y] = E[X] + E[Y]$$$ should only work if $$$X$$$ and $$$Y$$$ are "independent" (in some sense). Abstractly not so much, but looking at this problem it just seems so weird like, that can't possibly be true.

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      6 years ago, # ^ |
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      Not really. Expected of sum is sum of expected even if the random variables are not independent. This is used really often in problems, where you "break" the expected into a sum of dependent but computable sub-problems. I think that what you meant is that E(X and Y) = E(X) * E(Y) only and only if X and Y are independent probabilities. Check this for more info

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        6 years ago, # ^ |
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        No, I'm very well aware of how elementary probability theory works.

        All I'm saying is that intuitively, it does feel wrong.

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          5 years ago, # ^ |
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          When intuition gives incorrect answers, should we continue to trust it? or is there some way to fix intuition so that non-intuitive facts are intuitive?

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            5 years ago, # ^ |
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            Yes, if you work on something, solve problems etc for a while, your intuition will develop and correct its own mistakes.

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5 years ago, # |
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ban

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