Method of moments for a binomial with unknown count

Question

Each batch runs some fixed but unknown number of independent attempts $n$ , each succeeding with unknown probability $p$ , and you only ever record the total number of successes per batch. That total is Binomial $(n, p)$ . Across many batches the sample mean is $6$ and the sample variance is $2.4$ .

Use the method of moments to estimate $n$ and $p$ .

Accepted Answer

The two model moments. A Binomial(n, p) has \mathbb{E}[X] = np, \qquad \operatorname{Var}(X) = np(1 - p). Here the variance is smaller than the mean (because 1 - p < 1): that is "underdispersion," the opposite of the negative binomial's overdispersion. Solve for p first. Divide the variance by the mean: \frac{\operatorname{Var}(X)}{\mathbb{E}[X]} = 1 - p \implies \hat p = 1 - \frac{	ext{variance}}{	ext{mean}} = 1 - \frac{2.4}{6} = 1 - 0.4 = 0.6. Then solve for n. From the mean, n = 	ext{mean}/p: \hat n = \frac{6}{0.6} = 10. Check and a caveat. With n = 10 and p = 0.6: mean = 6 and variance…

Method of moments for a binomial with unknown count

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