Question

Consider a group ofn 4 people, numbered from l to n. For each pair (i, j) with ǐ关į person i and person J are friends, with probability p. Friendships are independent for different pairs. These n people are seated around a round table. For convenience, assume that the chairs are numbered from 1 to n, clockwise, with n located next to 1, and that person i seated in chair i. In particular, person 1 and person n are seatec next to each other If a person is friends with both people sitting next to him/her, we say this person is happy. Let H be the total number of happy people. We will find E H and Var(H) by carrying out a sequence of steps. Express your answers below in terms of p and/or n using standard notation or click on "STANDARD NOTATION button below). Remember to use"for multiplication and to include parentheses where necessary. We first work towards finding E H 1. Let li be a random variable indicating whether the person seated in chair z is happy or not (ie, Ii = 1 if person z is happy and li = 0 otherwise). Find E[L Fori 1,2,... ,n, ELL 2. Find E H) (Note: The notation a problem.) EH) means that a is defined to be E H. The simpler variable names will be used in the last question of this Since I1, 12,... , I, are not independent, the variance calculation is more involved. 3. For any k 11,2,...,n), find EE 4. For any i E ,2,...,n], and under the convention I+1-I, find E 41+1 5. Suppose thati j and that persons i and j are not seated next to each other. Find EI]. 6. Give an expression for Var(H), in terms of n, and the quantities a, b, c, d defined in earlier parts. Var(H)

Answer 1

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