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## 18.1 Pareto Distribution

### 18.1.1 Probability Density Function

If $$y_{\text{min}} \in \mathbb{R}^+$$ and $$\alpha \in \mathbb{R}^+$$, then for $$y \in \mathbb{R}^+$$ with $$y \geq y_{\text{min}}$$, $\text{Pareto}(y|y_{\text{min}},\alpha) = \frac{\displaystyle \alpha\,y_{\text{min}}^\alpha}{\displaystyle y^{\alpha+1}}.$

### 18.1.2 Sampling Statement

y ~ pareto(y_min, alpha)

Increment target log probability density with pareto_lupdf(y | y_min, alpha).

### 18.1.3 Stan Functions

real pareto_lpdf(reals y | reals y_min, reals alpha)
The log of the Pareto density of y given positive minimum value y_min and shape alpha

real pareto_lupdf(reals y | reals y_min, reals alpha)
The log of the Pareto density of y given positive minimum value y_min and shape alpha dropping constant additive terms

real pareto_cdf(reals y, reals y_min, reals alpha)
The Pareto cumulative distribution function of y given positive minimum value y_min and shape alpha

real pareto_lcdf(reals y | reals y_min, reals alpha)
The log of the Pareto cumulative distribution function of y given positive minimum value y_min and shape alpha

real pareto_lccdf(reals y | reals y_min, reals alpha)
The log of the Pareto complementary cumulative distribution function of y given positive minimum value y_min and shape alpha

R pareto_rng(reals y_min, reals alpha)
Generate a Pareto variate with positive minimum value y_min and shape alpha; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.