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The Number of Fixed Points in a Random Permutation


	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

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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