## 13.1 Bernoulli distribution

### 13.1.1 Probability mass function

If $$\theta \in [0,1]$$, then for $$y \in \{0,1\}$$, $\text{Bernoulli}(y~|~\theta) = \left\{ \begin{array}{ll} \theta & \text{if } y = 1, \text{ and} \\ 1 - \theta & \text{if } y = 0. \end{array} \right.$

### 13.1.2 Sampling statement

y ~ bernoulli(theta)

Increment target log probability density with bernoulli_lupmf(y | theta).
Available since 2.0

### 13.1.3 Stan Functions

real bernoulli_lpmf(ints y | reals theta)
The log Bernoulli probability mass of y given chance of success theta
Available since 2.12

real bernoulli_lupmf(ints y | reals theta)
The log Bernoulli probability mass of y given chance of success theta dropping constant additive terms
Available since 2.25

real bernoulli_cdf(ints y, reals theta)
The Bernoulli cumulative distribution function of y given chance of success theta
Available since 2.0

real bernoulli_lcdf(ints y | reals theta)
The log of the Bernoulli cumulative distribution function of y given chance of success theta
Available since 2.12

real bernoulli_lccdf(ints y | reals theta)
The log of the Bernoulli complementary cumulative distribution function of y given chance of success theta
Available since 2.12

R bernoulli_rng(reals theta)
Generate a Bernoulli variate with chance of success theta; 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