Beta-binomial Distribution - Moments and Properties

Moments and Properties

The first three raw moments are

$\begin{align} \mu_1 & =\frac{n\alpha}{\alpha+\beta} \\ \mu_2 & =\frac{n\alpha}{(\alpha+\beta)(1+\alpha+\beta)}\\ \mu_3 & =\frac{n\alpha}{(\alpha+\beta)(1+\alpha+\beta)(2+\alpha+\beta)} \end{align}$

and the kurtosis is

$\gamma_2 = \frac{(\alpha + \beta)^2 (1+\alpha+\beta)}{n \alpha \beta( \alpha + \beta + 2)(\alpha + \beta + 3)(\alpha + \beta + n) } \left.$

Letting we note, suggestively, that the mean can be written as

$\mu = \frac{n\alpha}{\alpha+\beta}=n\pi \!$

and the variance as

$\sigma^2 = \frac{n\alpha\beta(\alpha+\beta+n)}{(\alpha+\beta)^2(\alpha+\beta+1)} = n\pi(1-\pi) \frac{\alpha + \beta + n}{\alpha + \beta + 1} = n\pi(1-\pi) \!$

where is the pairwise correlation between the n Bernoulli draws and is called the over-dispersion parameter.

Read more about this topic: Beta-binomial Distribution

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