3. Test retaking. Each time a student takes an exam, their score is a independen

Question

3. Test retaking. Each time a student takes an exam, their score is a independent sample from a uniform distribution on [0, 100]. (a) If a student is allowed to take the exam twice, and average their two resulting scores, what will be their expected score? What if they are allowed to take the exam one hundred times and average their resulting scores? (b) An alternate policy allows a student to retake the exam once, but if they do, they must accept their second score as their final score. If a student only decides to retake the exam if their initial score is less than 50, what will their expected final score be!? (c) A very generous instructor allows their students to retake their exam as many times as they wish and then accept their most recent score. Describe a strategy that would ensure a student would get at least a 90 on the exam. What is the student's expected score in this case? Also, what is the average number of times the student must take the exam to get at least a 90?

Explanation / Answer

Here the their score is independent sample from a uniform distribution on [0,100]

f(x) = 1/100 ; 0 < x < 100

(a) Expected score if allowed to exam twice will be = E(x) = 50

expected score of they are allowed to eexam 100 times = E(x) = 50

as expected score will not change with respect of number of times they give the exam.

(b) Here expected final score is independent of his first score as these two are independent events. So, His expected score will be the same as earlier = E(x) = 50

This time also expected score of here exam will be 50.

(c) Here as the student will continuing repeating the exam atleast he get 90+ in the exam. so now the uniform distribution for final score limits it to 90 to 100.

so expected score in this case will be 95.

Here the strategy is to retake the exam as many times as they wish and in any exam if they get 90 or 90+ score. They should stop it and that would be counted as final score.

Here Pr(90 < x < 100) = (100 - 90)/100 = 0.1

so expected number of times the stdent must take the exam to get a least a 90 = 1/0.1 = 10

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