Estimate Grade Distribution Based on Performance of Each Question As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question.
To give a quick example of what I mean, suppose an exam has 2 questions worth 2 and 3 marks respectively:

Question 1: 20% receive 0 marks, 50% receive 1 mark, 30% receive 2
  marks.
Question 2: 10% receive 0 marks, 30% receive 1 mark, 40% receive 2
  marks,  20% receive 3 marks

Based on these individual distributions, I would like to find the overall distribution of marks (from 0 to 5).
My initial thoughts were:


*

*I could assume the performance on each question is independent of the other questions. However, I thought this unrealistic as a stronger student will likely get both Q1 and Q2 correct whereas a weaker student will likely get both wrong

*I could initially assign each student a 'strength' based on a normal (or other) distribution. Those at the right tail would be more likely to gain full marks and those at the left tail would be more likely to receive zero marks. However, such a model would not account for 'silly mistakes' or 'flukes'; i.e. even though a student's 'strength' is very large, there is still a probability of making a careless error or the probability a student guesses a question correct even though their 'strength' is very low.

*I also considered using models from my CT4 (actuarial stats) course, but I struggled with applying assumptions made for mortality investigations to my current problem.


Finally, I am familiar with the saying "all models are wrong but some are useful", but I would like to try get an accurate model if possible! Note: I do have actual numbers to verify any potential solution presented here and I will be happy to share them if need be :)
 A: This wont be a very satisfying answer for you as it will not be an accurate model, but it is maybe worth discussing.
I have no experience or specialised knowledge in this area, but right off the top of my head I have the following idea for a primitive model that would require lots of calculations and assume a strong dependence between the performances at the two questions. 
You divide up every question into three "quality sections", where the first section is the average mark of the lower 33% of the students, the second section is the middle 33% and the third section is the top 33%. We could then make the assumption that a student that belonged to section A in the first question would stay in section A in the second question with probability $1/2$ and would change to section B or section C with probability $1/4$, respectively. After assigning a student a section, the sections would have to be recalculated.
For rounding the marks: Let's say for example we have somebody that was in the top section at the first question, with an average mark of 1.8. I would then "roll the dice", meaning we assign 60% of the people in the top section 2 marks and $40$ of the people in the top section 1 mark. Of course, these assignments would make more revisions of the sections necessary.
I do not feel that I communicated the idea behind this very well or that it would be a very accurate model, but maybe I'll work out some code doing the calculations and we can see what numbers it would spit out. 
A: I don't think there's enough information in your model to work out a satisfactory answer.
If we draw up a table of scores like this:
 |0 1 2 3
-+-------
0|0 1 2 3
1|1 2 3 4
2|2 3 4 5

then I can populate it with the number of students like this (assuming only 10 students):
 |1 3 4 2
-+-------
2|1 1 - -
5|- 2 3 -
3|- - 1 2

Each column and row adds up to the number in the axes. Alternatively, I can populate like this:
 |1 3 4 2
-+-------
2|- - - 2
5|- 1 4 -
3|1 2 - -

However, I expect the real distribution to look more like this:
 |1 3 4 2
-+-------
2|- 1 1 -
5|1 1 2 1
3|- 1 1 1

by attempting to put a bell curve on each row and column. This would give you the following scores:


*

*0:  0%

*1: 20%

*2: 20%

*3: 30%

*4: 20%

*5: 10%


Alternatively, if you expect more correlation between the scores of both questions, it might look more like this:
 |1 3 4 2
-+-------
2|1 1 - -
5|- 2 2 1
3|- - 2 1

This is trying to both correlate the scores and use bell curves at the same time.
Final scores:


*

*0: 10%

*1: 10%

*2: 20%

*3: 20%

*4: 30%

*5: 10%

