Question

Suppose that 20% of the students who took a test from school A, and that the average of their scores on the teat was 80. Suppose that 30% of the students were from school B, and the average scores of the school is 76. Finally, suppose that 50% of the students are from school C , and that the average for yhat school is 84. If a student is selected at random from the entire group that took the test, what is the expected value of the elected students score? Use the properties of conditional expectaion to justify your answer.

Answer 1

Answer:

Expected Value: Expected value of any random variable X is given by: E(X)= Xxx Pr(x) Conditional Expectation: Consider two random variable X and Y with conditional distribution of X given Y as g(xly). The conditional expectation of X given y is: E(x|y) = f xg (x[y)dx It is known that: E[E(X|v)] = E(X)

Consider three classes A, B and C with 20%, 30% and 50% of the students in the school respectively. Average arithmetic score of these classes is 80, 76 and 84 respectively. Consider that a student is selected at random from the entire group. We are interested in finding expected arithmetic score of the student. Let X be the score of the selected student. Let events A, B and C represent that the student is selected form class A, class B or class C respectively. So, we have: Pr(A) = 0.2 Pr(B)=0.3 Pr(C)=0.5 The required expectation is: E(X)=E[E(X|Class)] = 80% Pr(A) +76x Pr(B) +84x Pr(C) = 80x0.2+76x0.3+84x0.5 = 80.8 So, the expected score of the randomly selected student is 80.8.

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