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.
