Each of n 2 people puts his or her name on a slip of paper n

Each of n 2 people puts his or her name on a slip of paper (no two have the same name). The slips of paper are shuffled in a hat, and then each person draws one (uniformly at random at each stage, without replacement). Find the standard deviation of the number of people who draw their own names.

Solution

E(X) = 2*1/(n-1) + 3/(n-1) + .....+ n/(n-1)

= 1/(n-1) * (2+3+...+n)

= 1/(n-1) * ((n^2+n)/2 -1)

E(X^2) = ^2*1/(n-1) + 3^2/(n-1) + .....+ n^2/(n-1)

= 1/(n-1) * (2^2+3^2+...+n^2)

= 1/(n-1) * ((n^2+n)(2n+1)/6 -1)

Var = E(x^2) - E(x)^2

= 1/(n-1) * ((n^2+n)(2n+1)/6 -1) - (1/(n-1) * ((n^2+n)/2 -1))^2

SD = sqrt(1/(n-1) * ((n^2+n)(2n+1)/6 -1) - (1/(n-1) * ((n^2+n)/2 -1))^2) Answer

Standard deviation =