Expected number of fixed points of permutation equals one

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Template:Symmetric group expected value statement


Suppose n is a natural number. Consider the uniform distribution on the symmetric group of degree n, i.e., the symmetric group on a set of size n. The expected number of fixed points for a permutation picked according to the uniform probability distribution equals 1.

See also probability distribution of number of fixed points of permutations.


We note that the number of permutations that fix a particular element i in the set is (n - 1)!, hence the probability that i is fixed is 1/n. By linearity of expectation, the expected number of fixed points is the sum, for each point, of the probability that it is fixed. This sum is n(1/n) = 1.

Note that expectation is linear even when the random variables in question are not independent.