# Bounded Continuous Distributions

The bounded continuous probabilities have support on a finite interval of real numbers.

## Uniform distribution

### Probability density function

If $$\alpha \in \mathbb{R}$$ and $$\beta \in (\alpha,\infty)$$, then for $$y \in [\alpha,\beta]$$, $\begin{equation*} \text{Uniform}(y|\alpha,\beta) = \frac{1}{\beta - \alpha} . \end{equation*}$

### Distribution statement

y ~ uniform(alpha, beta)

Increment target log probability density with uniform_lupdf(y | alpha, beta).

Available since 2.0

### Stan functions

real uniform_lpdf(reals y | reals alpha, reals beta)
The log of the uniform density of y given lower bound alpha and upper bound beta

Available since 2.12

real uniform_lupdf(reals y | reals alpha, reals beta)
The log of the uniform density of y given lower bound alpha and upper bound beta dropping constant additive terms

Available since 2.25

real uniform_cdf(reals y | reals alpha, reals beta)
The uniform cumulative distribution function of y given lower bound alpha and upper bound beta

Available since 2.0

real uniform_lcdf(reals y | reals alpha, reals beta)
The log of the uniform cumulative distribution function of y given lower bound alpha and upper bound beta

Available since 2.12

real uniform_lccdf(reals y | reals alpha, reals beta)
The log of the uniform complementary cumulative distribution function of y given lower bound alpha and upper bound beta

Available since 2.12

R uniform_rng(reals alpha, reals beta)
Generate a uniform variate with lower bound alpha and upper bound 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