Expected number of fixed points of permutation equals one

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.

Proof

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.