## 21.2 Pareto type 2 distribution

### 21.2.1 Probability density function

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

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

### 21.2.2 Sampling statement

y ~ pareto_type_2(mu, lambda, alpha)

Increment target log probability density with pareto_type_2_lupdf(y | mu, lambda, alpha).
Available since 2.5

### 21.2.3 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
Available since 2.18

real pareto_type_2_lupdf(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 dropping constant additive terms
Available since 2.25

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
Available since 2.5

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
Available since 2.18

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
Available since 2.18

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.
Available since 2.18