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.

Question

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.
1.Let Ii be a random variable indicating whether the person seated in chair i is happy or not (i.e., Ii=1 if person i is happy and Ii=0 otherwise). Find E[Ii] for i=1,2,..,n.
2.Find E[H]
3.Find E[Ik^2] for any k={1,2,...,n}
4.E[Ii.Ii+1] forany i {1,2,...,n}
5.Suppose that i!=j and that persons i and j are not seated next to each other. Find E[Ii.Ij]
6.Var(H) in terms of a=E[H],b=E[Ik^2],c=E[Ii.Ii+1], d=E[Ii.Ij] and n.

Answer 1

nobody cares to answer? :(