Consider a group of n ≥ 4 people, numbered from 1 to n . For each pair ( i , j ) with i ≠ j , 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 seated 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

We first work towards finding E [ H ] .

1. Let Ii be a random variable indicating whether the person seated in chair i is happy or not (i.e., I_i = 1 if person i is happy and I_i = 0 otherwise). Find E [ I_i ] . For i = 1 , 2 , … , n ,

2. Find E [ H ] . (Note: The notation a ≜ E [ H ] means that a is defined to be E [ H ] .

The simpler variable names will be used in the last question of this problem.)

3.Since I₁, I ₂ , … , I_n are not independent, the variance calculation is more involved. For any k ∈ { 1 , 2 , … , n } , find E [ I_k²] .

4.For any i ∈ { 1 , 2 , … , n } , and under the convention I n + 1 = I 1 , find E [ I _i I _i+1 ]

5.Suppose that i ≠ j and that persons i and j are not seated next to each other. Find E [ I _i I _j ] .

6. Give an expression for Var ( H ) , in terms of n , and the quantities a , b , c , d defined in earlier parts.