# Beta as distribution of proportions (or as continuous Binomial)

Beta distribution is related to binomial being also the distribution for order statistics. Probability mass function of binomial distribution is

$$f(k) = {n \choose k} p^k (1-p) ^{n-k} \tag{1}$$

Probability density function of beta distribution is

$$g(p) = \frac{1}{\mathrm{B}(\alpha, \beta)} p^{\alpha-1} (1-p)^{\beta-1} \tag{2}$$

we can rewrite $n \choose k$ in (1) as

$$\frac{1}{(n+1) \mathrm{B}(k+1, n-k+1)}$$

if we substitute $k+1 = \alpha$ and $n-k+1 = \beta$ then (1) becomes

$$\frac{1}{(n+1) \mathrm{B}(\alpha, \beta)} p^{\alpha-1} (1-p)^{\beta-1}$$

So basically, beta is a distribution of $k/n$ proportions in $n$ trials where average proportion is denoted as $\mu$

$$\frac{1}{\mathrm{B}(n\mu+1, n(1-\mu)+1)} p^{n\mu} (1-p)^{n(1-\mu)} \tag{3}$$

Are you familiar with any references or examples of such usage of beta? Most literature on statistic analysis with proportions (that I found) seems to describe only binomial distribution and beta-binomial Bayesian model rather than dealing directly with beta.

• crcpress.com/… Feb 5, 2016 at 11:53
• @NickCox right, beta regression! I knew that I saw it somewhere! There is even a paper suggesting the same parametrizartion as mine: ime.usp.br/~sferrari/beta.pdf If you would like to provide an answer I'd be happy to accept it. Otherwise I'd answer it myself.
– Tim
Feb 5, 2016 at 12:09

## 2 Answers

The beta as a distribution for variables that are, or are like, proportions is a a popular playground in several fields of statistical science. Beta regression is a major focus of this text and monograph.

As that book and other literature exemplify in detail, that still leaves scope for discussion in general and in particular over the merits of such models as compared with generalised linear models using a binomial family and (usually) robust-sandwich-Huber-White flavour, let alone linear probability models.

Adding to Nick's answer, such parametrization as described in my question is also mentioned in vignette to betareg package and by Ferrari and Cribari-Neto (2004) who also proposed it in their paper about Beta regression. Ferrari and Cribari-Neto (2004) describe the parameters as $\mu$ for mean and $\phi$ for precision (i.e. as it grows, it decreases variance).

Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.