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18.5 Student-t distribution

18.5.1 Probability density function

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

18.5.2 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

18.5.3 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