T is a continuous variable uniformly distributed between 0 and 100. t, s1,s2 and s3 are randomly chosen numbers from this distribution where t is called the target number. The value of t, s1 and s2 are not disclosed whereas the value of s3 is publicly known
It is also revealed that s1 is less than or equal to t, s2 is greater than t and less than or equal to 75. Finally, s3 is greater than t.
What is expected value of t conditional on the given information above?
Note that the expected value of t (target number) should be dependent on only s3 since the value of s1 and s2 are not known.
I have the solution for the case where two random numbers are drawn to compare with the target number, t. In that case, lets call the random numbers as s1 and s2 where only the value of s2 is disclosed (s1 is unknown).
It is revealed that s1 is less than or equal to t.
If s2 is greater than or equal to target number t, the expected value of t becomes 2*s2/3. The solution can be found via double integral
$\int^{s2}_0$$\int^y_0$$(s2-x)dxdy$/$\int^{s2}_0$$\int^y_0$$dxdy$=$2*s2/3$
If s2 is less than target number t, the expected value of t becomes (2/3)[$(100+s2)^2$-100s2]/(100+s2). The solution can be found via double integral or the gravity center of the area.