# Continuous Distributions on [0, 1]

The continuous distributions with outcomes in the interval $$[0,1]$$ are used to characterized bounded quantities, including probabilities.

## Beta distribution

### Probability density function

If $$\alpha \in \mathbb{R}^+$$ and $$\beta \in \mathbb{R}^+$$, then for $$\theta \in (0,1)$$, $\begin{equation*} \text{Beta}(\theta|\alpha,\beta) = \frac{1}{\mathrm{B}(\alpha,\beta)} \, \theta^{\alpha - 1} \, (1 - \theta)^{\beta - 1} , \end{equation*}$ 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, $$\alpha, \beta > 0$$.

### Distribution statement

theta ~ beta(alpha, beta)

Increment target log probability density with beta_lupdf(theta | alpha, beta).

Available since 2.0

### 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

Available since 2.12

real beta_lupdf(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 dropping constant additive terms

Available since 2.25

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

Available since 2.0

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

Available since 2.12

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

Available since 2.12

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.

Available since 2.18

## Beta proportion distribution

### Probability density function

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

### Distribution statement

theta ~ beta_proportion(mu, kappa)

Increment target log probability density with beta_proportion_lupdf(theta | mu, kappa).

Available since 2.19

### 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

Available since 2.19

real beta_proportion_lupdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion density of theta in $$(0,1)$$ given mean mu and precision kappa dropping constant additive terms

Available since 2.25

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

Available since 2.18

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

Available since 2.18

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

Available since 2.18