Applications of Beta Distribution What properties of random variable leads to modeling it with Beta Distribution?
Context: If you ask the same question about Bernouli distribution, the answer would be the distribution of a random variable that has only two outcomes and the probability of one outcome is fixed at $p$.
Now how can we answer a similar question about Beta distribution?
 A: One of the most important uses of beta distributions is in Bayesian
statistical analysis of binomial data. Beta distributions are used for
prior and posterior distributions. I will give an elementary example below.
In Bayesian statistics a beta distribution is used as a model for the
binomial success probability, which I will denote as $\theta$ (instead of $p$).
[Bayesian analysis treats $\theta$ as a random variable, rather than as
an unknown fixed parameter.]
Suppose you are wondering whether Proposition A will get a majority of
the vote at the next election. The circumstances are that Prop A had to
have some support in order to get on the ballot. Also a similar proposition
passed with about 65% of the vote at the last election. That may be
good or bad: perhaps good because it may show voters favor such propositions;
perhaps bad because such propositions require a slight increase in property
taxes and it may be too soon to ask again.
Everything considered, you think Prop A may be favored by slightly more
than half of the voters, but that the election is likely to be close. 

Because
  $0 \le \theta \le 1$ a beta distribution seems appropriate because
  beta distributions have support $[0,1].$ 

There is no precisely correct
prior distribution for $\theta,$ because it should reflect your personal
opinion. 
Suppose you choose the prior distribution $\mathsf{Beta}(\alpha_0 = 330, \beta_0=270).$
The 'kernel' of its PDF (omitting the constant of integration) is 
$f(\theta) \propto \theta^{\alpha - 1}(1-\theta)^{\beta-1},$ where the 
symbol $\propto$ indicates omission of the constant. It has mean 0.55, mode 0.5502, and median 0.55006. Also, 
$P(0.51 < \theta < 0.59) \approx 0.95,$ as computed in R statistical software.
qbeta(.5, 330, 270)                         #  'qbeta' is inverse CDF
## 0.5500556
pbeta(.59, 330, 270) - pbeta(.51, 330, 270) #  'pbeta' is CDF
## 0.9513758

Now suppose that a well-run poll of $n = 1000$ randomly chosen likely voters
shows $x = 620$ in favor of Prop A and $n - x  = 380$ opposed. The likelihood
function corresponding to these results is 
$f(x|\theta) \propto \theta^x(1-\theta)^{n-x}.$
Bayes Theorem states that the posterior distribution $f(\theta|x)$ is found
by multiplying the prior and likelihood functions:
$$f(\theta|x) \propto \theta^{\alpha_0 - 1}(1-\theta)^{\beta_0-1}
\times \theta^x(1-\theta)^{n-x}.$$
In our example, $$f(\theta|x) = 
\theta^{\alpha_0 + x -1}(1-\theta)^{\beta_0 + n - x -1} = 
\theta^{\alpha_n - 1}(1-\theta)^{\beta_n - 1},$$
where $\alpha_n = 950$ and $\beta_n = 650.$
We recognize $f(\theta|x)$ as the kernel of $\mathsf{Beta}(950, 650).$

We say that the beta prior and the binomial likelihood are 'conjugate' 
  (mathematically compatible). [Without conjugacy we would not be able to
  recognize the posterior distribution so easily, and we would have to
  use a different form a Bayes' Theorem in which a denominator might
  need to be integrated by numerical methods.]

Finally, a 95% posterior probability interval for $\theta$ is $(0.570, 0.618).$
qbeta(c(.025,.975), 950, 650)
## 0.5695848 0.6176932

A melding of the information in the prior and the likelihood has given a
somewhat more optimistic estimate of the chances Prop A will pass, than
did our prior distribution.

Notes: (1) If we had used the noninformative prior distribution $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1),$ then the posterior distribution would have
been $\mathsf{Beta}(621,381)$ with a 95% posterior probability interval
$(0.589, 0.650).$ This is numerically the same (to three places) as a frequentist Agresti-style 95% confidence interval for $\theta$. However, Bayesian and frequentist interval estimates have somewhat different interpretations.
(2) This example is similar to Example 8.1 in Suess (2010).
