A round table with 20 seats is chosen for dinner for a party with ten couples. They enter the room and sit at random chairs at the table. Let Y be the number of couples that sit together. We want to compute E[Y] and Var[Y].
(a) Define binary variable Xi = 1 if and only if Mr.i and Ms.i sit next together. Compute P[Xi = 1].
(b) What are E[Xi] and Var[Xi].
(c) Express Y in terms of Xi’s.
(d) What is E[Y]?