My question illustrated in a problem:
After giving and grading a statistics exam, the professor found the mean and standard deviation of points students got for each problem. One of the professors students had to miss the exam, and scheduled to take it later. With the calculated distributions only, what is the probability that the late student gets at least X% on the test?
The number of problems and percent cutoff are not predefined, and the points, mean, and standard deviation of each problem are independent to other problems.
You can assume each distribution is purely gaussian, if that helps.
For whatever reason, say that the professor doesn't have access to the individual students' data anymore, so he/she couldn't just count the number of students with a certain overall score, make a new distribution of overall scores, or any other workaround. The professor only has a set of independent gaussian distributions.
In short: summing together one random value per independent gaussian distribution, what is the probability of getting over a certain total value?