The Number of Fixed Points in a Random Permutation

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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
Snapshots
Details
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.
Permanent Citation