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## 20.1 Pareto distribution

### 20.1.1 Probability density function

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

### 20.1.2 Sampling statement

y ~ pareto(y_min, alpha)

Increment target log probability density with pareto_lupdf(y | y_min, alpha).
Available since 2.0

### 20.1.3 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
Available since 2.12

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
Available since 2.25

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
Available since 2.0

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
Available since 2.12

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
Available since 2.12

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.
Available since 2.18