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.
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.