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## 17.2 Pareto Type 2 Distribution

### 17.2.1 Probability Density Function

If $$\mu \in \mathbb{R}$$, $$\lambda \in \mathbb{R}^+$$, and $$\alpha \in \mathbb{R}^+$$, then for $$y \geq \mu$$, $\mathrm{Pareto\_Type\_2}(y|\mu,\lambda,\alpha) = \ \frac{\alpha}{\lambda} \, \left( 1+\frac{y-\mu}{\lambda} \right)^{-(\alpha+1)} \! .$

Note that the Lomax distribution is a Pareto Type 2 distribution with $$\mu=0$$.

### 17.2.2 Sampling Statement

y ~ pareto_type_2(mu, lambda, alpha)

Increment target log probability density with pareto_type_2_lpdf(y | mu, lambda, alpha) dropping constant additive terms.

### 17.2.3 Stan Functions

real pareto_type_2_lpdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 density of y given location mu, scale lambda, and shape alpha

real pareto_type_2_cdf(reals y, reals mu, reals lambda, reals alpha)
The Pareto Type 2 cumulative distribution function of y given location mu, scale lambda, and shape alpha

real pareto_type_2_lcdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 cumulative distribution function of y given location mu, scale lambda, and shape alpha

real pareto_type_2_lccdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 complementary cumulative distribution function of y given location mu, scale lambda, and shape alpha

R pareto_type_2_rng(reals mu, reals lambda, reals alpha)
Generate a Pareto Type 2 variate with location mu, scale lambda, 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.