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I am working on a problem like this: Suppose that there are $N$ students in the class. $n_1$ students take course A, $n_2$ take course B and $n_{12}$ take courses both A and B. What is the MLE of $N$?

I did in the following way: assume that $p_1$ is the probability that a student take Course A, and $p_2$ is the probability that a student take Course B, then $n_1\sim Bin(N,p_1)$ and $n_2\sim Bin(N,p_2)$ so that in case $N$ is known we will get $\hat p_1=n_1/N$ and $\hat p_2=n_2/N$.

Similarly for $\hat p_{12}=n_{12}/N$.

Notice that $\hat p_{12}=\hat p_1 \hat p_2$, we will have the following equation $(n_1/N)(n_2/N)=n_{12}/N$, from which we get $\hat N=n_1 n_2/n_{12}$.

But I am wondering whether this reasoning is correct as I assumed $N$ is known first.

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    You are assuming that taking course A and taking course B are independent. That is a strong assumption and may not be justified.2011-12-12
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    This problem seems to be underdetermined. Suppose $n_0$ is the number of students enrolled in neither $A$, nor $B$. Then $N = n_0 + n_1 + n_2 - n_{12}$. It seems the information on the possible distribution of $n_0$ is missing.2011-12-12
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    @Henry, what about we do have the assumption that they are independent?2011-12-13
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    @Dilip Sarwate, it seems that I cannot download the link you gave..2011-12-13
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    Sorry, those should have had [homework](http://courses.engr.illinois.edu/ece313/spring05/homework/PS04.pdf) instead of problems in the file directory. Try [these ones](http://courses.engr.illinois.edu/ece313/spring05/homework/HW04.pdf) instead.2011-12-13
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    @Sasha, as mentioned by Sasha, could we just set the problem by multinomial distribution with $(N,p_1p_2,p_1(1-p_2),p_2(1-p_1),(1-p_1)(1-p_2))$? Then we have observations as $n_1$, $n_2$, $n_12$2011-12-13

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