Question

Consider a class with 3 sections denoted as A, B, C. Let the number of students in section A be 10, in section B be 20, and in section C be 15. Let the score of uniformly chosen student be denoted by random variable X. The student’s section be denoted by the random variable Y where Y = 1 if the student is from section A, Y = 2 if the student is from section B and Y = 3 if the student is from section C. In a certain test, the mean score of the students in section A is 90, in section B is 60 and in section C is 75. The variance of the scores in section A, B and C is 5, 10 and 15 respectively.

Answer the following questions:

1. Calculate the conditional expectation E[X|Y].

2. Calculate the mean score of entire class, E[X].

3. Calculate the conditional variance var(X|Y).

4. Calculate the total variance of entire class var(X).

Answer 1

1

X - the score of uniformly chosen student

Y - The student’s section

Y = 1 if the student is from section A, Y = 2 if the student is from section B and Y = 3 if the student is from section C

.

the mean score of the students in section A is 90, in section B is 60 and in section C is 75

the conditional expectation is given by,

2.

the mean of the entire class is given by,

3.

The variance of the scores in section A, B and C is 5, 10 and 15 respectively

the conditional variance is given by,

4.

The overall variance of X is given by,