Asked by stud81
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)= ?
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)= ?
Answers
Answered by
stud81
I have posted 7 Probability questions and i am willing to pay $ for the solution.
Answered by
Anonymous
can anyone please answer the question?
Answered by
stud81
Hi Anonymous,
Can you share answers to any of the other questions ....?
Can you share answers to any of the other questions ....?
Answered by
Anonymous
E[Ii]= p^2
E[H]-n.p^2
thats all i can calculate. stuck after this.
E[H]-n.p^2
thats all i can calculate. stuck after this.
Answered by
who cares
you are supposed to try and find your own solutions for the edx probability course
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.