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 $\theta \in (0,1)$, $$\text{Beta}(\theta |\alpha ,\beta )=\frac{1}{\mathrm{B}(\alpha ,\beta )}{\theta }^{\alpha -1}(1-\theta {)}^{\beta -1},$$ 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 $-\mathrm{\infty }$. 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 $\theta \in (0,1)$, $$\mathrm{Beta}\mathrm{\_}\mathrm{Proportion}(\theta |\mu ,\kappa )=\frac{1}{\mathrm{B}(\mu \kappa ,(1-\mu )\kappa )}{\theta }^{\mu \kappa -1}(1-\theta {)}^{(1-\mu )\kappa -1},$$ 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 $-\mathrm{\infty }$. 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