15.5 Student-T Distribution

15.5.1 Probability Density Function

If \(\nu \in \mathbb{R}^+\), \(\mu \in \mathbb{R}\), and \(\sigma \in \mathbb{R}^+\), then for \(y \in \mathbb{R}\), \[ \text{StudentT}(y|\nu,\mu,\sigma) = \frac{\Gamma\left((\nu + 1)/2\right)} {\Gamma(\nu/2)} \ \frac{1}{\sqrt{\nu \pi} \ \sigma} \ \left( 1 + \frac{1}{\nu} \left(\frac{y - \mu}{\sigma}\right)^2 \right)^{-(\nu + 1)/2} \! . \]

15.5.2 Sampling Statement

y ~ student_t(nu, mu, sigma)

Increment target log probability density with student_t_lpdf( y | nu, mu, sigma) dropping constant additive terms.

15.5.3 Stan Functions

real student_t_lpdf(reals y | reals nu, reals mu, reals sigma)
The log of the Student-\(t\) density of y given degrees of freedom nu, location mu, and scale sigma

real student_t_cdf(reals y, reals nu, reals mu, reals sigma)
The Student-\(t\) cumulative distribution function of y given degrees of freedom nu, location mu, and scale sigma

real student_t_lcdf(reals y | reals nu, reals mu, reals sigma)
The log of the Student-\(t\) cumulative distribution function of y given degrees of freedom nu, location mu, and scale sigma

real student_t_lccdf(reals y | reals nu, reals mu, reals sigma)
The log of the Student-\(t\) complementary cumulative distribution function of y given degrees of freedom nu, location mu, and scale sigma

R student_t_rng(reals nu, reals mu, reals sigma)
Generate a Student-\(t\) variate with degrees of freedom nu, location mu, and scale sigma; may only be used in generated quantities block. For a description of argument and return types, see section 10.8.3.