The Number of Fixed Points in a Random Permutation

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send