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## 11.1 Bernoulli Distribution

### 11.1.1 Probability Mass Function

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

### 11.1.2 Sampling Statement

y ~ bernoulli(theta)

Increment target log probability density with bernoulli_lpmf( y | theta) dropping constant additive terms.

### 11.1.3 Stan Functions

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

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

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

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

R bernoulli_rng(reals theta)
Generate a Bernoulli variate with chance of success theta; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.