## 19.5 Student-t distribution

### 19.5.1 Probability density function

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

### 19.5.2 Sampling statement

y ~ student_t(nu, mu, sigma)

Increment target log probability density with student_t_lupdf(y | nu, mu, sigma).
Available since 2.0

### 19.5.3 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
Available since 2.12

real student_t_lupdf(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 dropping constant additive terms
Available since 2.25

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
Available since 2.0

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
Available since 2.12

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
Available since 2.12

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 transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.
Available since 2.18