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Suppose we have a random variable $X = [x_1, x_2, ..., x_m]$, that is distributed $Binomial(n,p)$, with known $n$ and unknown $p$.

Now, assume we want to estimate $p$. Usually, textbooks and articles online give that the MLE of $p$ is $\frac{\sum_{i=1}^{m}x_i}{n}$. However, isn't it correct only when $m=1$, or in other words, when we end up having merely a $Bernoulli$ distribution?

If so, wouldn't it be more precise to say that the MLE of $p$ is actually $\frac{\sum_{i=1}^{m}x_i}{mn}$?

R.Astur
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You are right, while even the credible sources sometimes claim differently, the correct formula is

$$ \hat p = \frac{1}{mn} \sum_{i=1}^m x_i = \overbrace{\frac{1}{m} \sum_{i=1}^m}^\text{we have m trials} \overbrace{\frac{x_i}{n}}^{\substack{\text{proportion of successes} \\ \text{in single Bernoulli trial}}} $$

since if you calculated ordinary arithmetic mean of $X$, you'd get average number of successes in $n$ trials, i.e. $E(X) = n \hat p$.

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Tim
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  • That's exactly how I thought, however I saw so many books and sources giving the $Bernoulli$ formula when talking about $Binomial$ that I got to the point of questioning myself. Thanks! – R.Astur Oct 17 '16 at 18:37