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## 19.2 Beta Proportion Distribution

### 19.2.1 Probability Density Function

If $$\mu \in (0, 1)$$ and $$\kappa \in \mathbb{R}^+$$, then for $$\theta \in (0,1)$$, $\mathrm{Beta\_Proportion}(\theta|\mu,\kappa) = \frac{1}{\mathrm{B}(\mu \kappa, (1 - \mu) \kappa)} \, \theta^{\mu\kappa - 1} \, (1 - \theta)^{(1 - \mu)\kappa- 1} ,$ 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 $$\mu \in (0, 1)$$ and strictly positive parameter, $$\kappa > 0$$.

### 19.2.2 Sampling Statement

theta ~ beta_proportion(mu, kappa)

Increment target log probability density with beta_proportion_lpdf( theta | mu, kappa) dropping constant additive terms.

### 19.2.3 Stan Functions

real beta_proportion_lpdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion density of theta in $$(0,1)$$ given mean mu and precision kappa

real beta_proportion_lcdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion cumulative distribution function of theta in $$(0,1)$$ given mean mu and precision kappa

real beta_proportion_lccdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion complementary cumulative distribution function of theta in $$(0,1)$$ given mean mu and precision kappa

R beta_proportion_rng(reals mu, reals kappa)
Generate a beta_proportion variate with mean mu and precision kappa; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.