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## 19.1 Beta Distribution

### 19.1.1 Probability Density Function

If $$\alpha \in \mathbb{R}^+$$ and $$\beta \in \mathbb{R}^+$$, then for $$\theta \in (0,1)$$, $\text{Beta}(\theta|\alpha,\beta) = \frac{1}{\mathrm{B}(\alpha,\beta)} \, \theta^{\alpha - 1} \, (1 - \theta)^{\beta - 1} ,$ where the beta function $$\mathrm{B}()$$ is as defined in section combinatorial functions.

Warning: If $$\theta = 0$$ or $$\theta = 1$$, then the probability is 0 and the log probability is $$-\infty$$. Similarly, the distribution requires strictly positive parameters, $$\alpha, \beta > 0$$.

### 19.1.2 Sampling Statement

theta ~ beta(alpha, beta)

Increment target log probability density with beta_lpdf( theta | alpha, beta) dropping constant additive terms.

### 19.1.3 Stan Functions

real beta_lpdf(reals theta | reals alpha, reals beta)
The log of the beta density of theta in $$[0,1]$$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

real beta_cdf(reals theta, reals alpha, reals beta)
The beta cumulative distribution function of theta in $$[0,1]$$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

real beta_lcdf(reals theta | reals alpha, reals beta)
The log of the beta cumulative distribution function of theta in $$[0,1]$$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

real beta_lccdf(reals theta | reals alpha, reals beta)
The log of the beta complementary cumulative distribution function of theta in $$[0,1]$$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

R beta_rng(reals alpha, reals beta)
Generate a beta variate with positive prior successes (plus one) alpha and prior failures (plus one) beta; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.