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A classmate sent me this question:

A high school student had the option to attend 5 AP classes. If we assume the probability of signing up for each class is constant, then the amount of AP courses a student would take is a random variable distributed as:

X ~ Binomial(n=5, p).

If we have a sample of 120 high school students denoted $X_1, ..., X_{120}$, what would be the maximum likelihood estimate of $p$?

We have this formula from our notes:

$L(p)=∏^n_{i=1}p^{x_i}(1−p)^{1−x_i}$.

However, he suggested maximizing the function $p^{x}(1−p)^{n−x}$ [here $x = \sum \limits_{i=1}^n x_i$], with $n=5$. Then we simply obtain that the maximum likelihood estimate as $p=\frac{x}{5}$. I feel that either I or the problem is missing some key information. Is this correct?

Shanks
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KitCarpson
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    You need a bit more notation; have to be careful to distinguish your number of available classes from the number of students. Note that you have 120 binomial observations -- your likelihood is just for one student. Do they all have the same $p$? If so, you need to incorporate all of the observations into the likelihood (see the definition of likelihood). – Glen_b May 09 '18 at 06:12

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