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## 16.6 Cauchy distribution

### 16.6.1 Probability density function

If $$\mu \in \mathbb{R}$$ and $$\sigma \in \mathbb{R}^+$$, then for $$y \in \mathbb{R}$$, $\text{Cauchy}(y|\mu,\sigma) = \frac{1}{\pi \sigma} \ \frac{1}{1 + \left((y - \mu)/\sigma\right)^2} .$

### 16.6.2 Sampling statement

y ~ cauchy(mu, sigma)

Increment target log probability density with cauchy_lupdf(y | mu, sigma).

### 16.6.3 Stan functions

real cauchy_lpdf(reals y | reals mu, reals sigma)
The log of the Cauchy density of y given location mu and scale sigma

real cauchy_lupdf(reals y | reals mu, reals sigma)
The log of the Cauchy density of y given location mu and scale sigma dropping constant additive terms

real cauchy_cdf(reals y, reals mu, reals sigma)
The Cauchy cumulative distribution function of y given location mu and scale sigma

real cauchy_lcdf(reals y | reals mu, reals sigma)
The log of the Cauchy cumulative distribution function of y given location mu and scale sigma

real cauchy_lccdf(reals y | reals mu, reals sigma)
The log of the Cauchy complementary cumulative distribution function of y given location mu and scale sigma

R cauchy_rng(reals mu, reals sigma)
Generate a Cauchy variate with location mu and scale sigma; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.