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 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 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=0otherwise). Find E[Ii].
For i=1,2,…,n,
E[Ii]= ?
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.)
a≜E[H]= ?
Since I1,I2,…,In are not independent, the variance calculation is more involved.
3. For any k∈{1,2,…,n}, find E[Ik^2].
b≜E[Ik^2]= ?
4. For any i∈{1,2,…,n}, and under the convention In+1=I1, find E[IiIi+1].
c≜E[Ii Ii+1]= ?
here: (i + 1 is a subscript)
5. Suppose that i≠j and that persons i and j are not seated next to each other. Find E[IiIj].
d≜E[IiIj]= ?
6. Give an expression for Var(H), in terms of n, and the quantities a,b,c,d defined in earlier parts.
Var(H)= ?
5 answers
Can you share answers to any of the other questions ....?
E[H]-n.p^2
thats all i can calculate. stuck after this.
1 answer