Suppose that n people have their hats returned at random Let Xi 1 if the ith person gets his or her own hat back and 0 otherwise Let Sn n i 1 Xi Then Sn is the total number of people who get their own hats back Show that (a) E (X2 i) 1 n (b) E (Xi Xj) 1 n (n 1) for i 6 j (c) E (S2 n) 2 (using (a) a...

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Suppose that n people have their hats returned at random. Let Xi = 1 if the ith

Question:

Suppose that n people have their hats returned at random. Let Xi = 1 if the ith person gets his or her own hat back and 0 otherwise. Let Sn = Σn i=1 Xi. Then Sn is the total number of people who get their own hats back. Show that
(a) E (X2 i) = 1/n.
(b) E (Xi • Xj) = 1/n (n − 1) for i 6≠ j.
(c) E (S2/n) = 2 (using (a) and (b)).
(d) V (Sn) = 1.