The Number of Fixed Points in a Random Permutation
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For a random permutation of 
, let 
be the random variable that counts the number of digits that remain in their original position. This Demonstration allows you to compare the relative frequencies of obtained in a sample of size 400 with the exact and approximate distributions of . It also gives the sample mean and standard deviation.
Contributed by: Elcio Lebensztayn (May 2008)
Open content licensed under CC BY-NC-SA
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This is the so-called matching problem, in which 
individuals mix their hats up and then randomly make a selection. The random variable is the number of individuals that select their own hat. The permutations that lead to 
are called derangements. The distribution of is given by 
, 
. Both the expectation and the variance of equal 1, regardless of the value of . As goes to infinity, the distribution of converges to the Poisson distribution with parameter 1.
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