Consider a group of n>=4 people, numbered from 1 to n. For each pair (i,j) with $i \not =j$ 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 to 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. Express your answers in terms of p and/or n
It says Lets $I_i$, be a random variable indicating whether the person seated in chair i is happy or not ($I_i$ = 1 if person i is happy and 0 otherwise). Find $E[I_i]$ for i = 1,2,...,n
I calculated $E(I_i)=p^2$ and $E[H]=n*p^2.$ I'm stuck on the parts where I have to calculate the following
- $E[I_k^2]$ for any k in set 1,2,..n.
2.$E[I_iJ_j]$ for $i \not =j$
3.under the convention $I_n+_1=I_1$, find $E[I_iI_n+_1]$
4.Var(H)