# 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/… – Nick Cox Feb 5 '16 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 '16 at 12:09

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).