A student has the opportunity to take a test at most thrice. The student knows that each time he takes the test, his score is independent random draw from uniform distribution on interval [0,100]. After learning his score on a test, the student can either stop and accept it as his official score , or he can discard the result and retake the test. If the student rejects his score twice and takes the test third time, that score will be his official score. If his objective is to maximise his expected official score, the student will decide to be retested after the very first test if and only if his score is less than a)50)62.5 c)87.5
I tried to define c1 and c2 as the cutoffs for test 1 and 2.So if he scores less than c1 on test 1 ,he will ask for a retest.Similarily for test 2. If the X is the final score than X= X1(if X1>m1),X2(if X1 less than m1 and X2 greater than m2),=X3 otherwise..i am not sure how to find the expectation of this process
If the student takes the third test then his expected mark is $50$, as this is the mean of the uniform distribution on $[0,100]$.
Now suppose the student takes the second test. There is a $\frac{1}{2}$ chance that he will score $50$ or more. In this case he will not take the third test (because if he did, his expected mark would be lower, or at best the same). So his final mark will be what he got for test $2$. As this case is a uniform distribution on $[50,100]$ the expectation is $75$.
On the other hand, there is a $\frac{1}{2}$ chance that he will get less than $50$ on the second test. In this case he will take the third test and his expected mark is $50$.
Putting it all together, if the student takes the second test, his expected score is $$\frac{1}{2}75+\frac{1}{2}50=62.5\ .$$ So, if he receives the first test mark and it is this much or better, then he should stop and and not take any more tests.