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# 890 Paper
# 890 Score Report

Dear Team #890,
 
Your paper is neatly organized with all necessary parts which leaves us a very good first impression. Both of your models use theoretical approach to the problems and derive the solution from pure theory, clearly demonstrating your mathematical knowledge in probability distributions and spherical geometry.
 
However, there are several problems that occur in both of your models.
 
First, preferably, sensitivity analysis should indicate how the two possible sources of errors will affect the outcome of your model. Will your model be robust enough to handle them? You have good sets of assumptions, but you also need corresponding justifications. Second, more details about methods and results are needed in the summary. Your conclusion is also too brief and needs to be expanded. Third, don’t forget to put citation in text. It is extremely important to state all your variables at first, which makes your paper a lot easier to understand. For example, you need to specify t is the given set of possible grades, T = t means that grade “t” is earned on test T. Is n = # of questions on a given test?

There are a few improvements that could be made with paper 1. Firstly, plots and graphs would make it much easier for the readers to visualize the distributions. One portion of your paper says that ability is random and uniformly distributed, but elsewhere says that ability is based on previous tests (must stay consistent). You need to prove exactly why f(A,D) is a bernoulli trial; currently, your explanation is not specific enough. You should try to explain why you wrote eq 1-5. Another error is that  all the Tn should be Tm. Also, in the last step where you substitute it, the sum of l = 1 to m should actually be the sum of m = 0 to n. In addition, according to eq.6, all the sum of m = 0 to n should start with m = 1 instead. Again, you must stay consistent with your variables. Why are you switching from t to capitalized T starting with eq. 7? If you want to say there is a probability of 1/n that T = t because the number of possible grades is the size of unit t, then S = s should also be a constant. Pr(S=s|T=Tl) should be the same as Pr(T=Tl|S=s). The only thing done here was replacing the variable n with m in equation. You stated that you are trying to prove that the sum of individual tests should follow a poisson binomial distribution, but in eq. 11, you are using the grade received on each test, and that is not the same as what you are intending to say according to your eq. 6. If you are trying to say that if there are x people receiving the same grade on the test, there will be a probability of x/m that Pr(T = t), please state so. If you did a better job stating your variables, it would be much easier to understand all of the equations.
 
You also need to correctly express the problem in mathematical terms. According to the first paragraph of 2.2.5, it seems that you want the difficulties of problems to be randomly distributed. If so, you must clearly state this in both conclusion and summary. Overall, a much clearer explanation should be given for eq. 6-11. 2.2.5 is sufficient if better explanation is given in 2.2.4. Your idea that the distribution of scores should be the sum of individual scores divided by number of tests is really good. However, your proof does not appear to be sufficient enough. Other than that, this is a pretty well written and thorough paper. 
 
There are also a few improvements that could be made with paper 2. Firstly, you need to add units to all necessary figures and clearly indicate where each of the numbers that you used in your paper comes from. For instance, show how you obtained 88.96 degrees and 135 degrees; if you can’t derive the whole formula, at least state the variables you used to derive these values. Otherwise they seem to come out of nowhere. Showing work is essential for problems like these; how do you know the height of the orbit needs to be 330,000km? Also, why are the satellites moving slightly faster than the earth? You must explain each detail about your final distribution of satellites to help the judges understand your paper. In addition, in equation 27, angle B is not 67.522 and area of intersection is not what you claimed according to your formula (eq. 34).  Other than these errors, the rest of your work is correct. Finally, it is essential to incorporate some sort of explanations about the two figures in the appendix.
 
Best,
Association of Computational and Mathematical Modeling