dist_student_t: The (non-central) location-scale Student t Distribution

In distributional: Vectorised Probability Distributions

The (non-central) location-scale Student t Distribution

Description

The Student's T distribution is closely related to the Normal() distribution, but has heavier tails. As \nu increases to \infty, the Student's T converges to a Normal. The T distribution appears repeatedly throughout classic frequentist hypothesis testing when comparing group means.

Usage

dist_student_t(df, mu = 0, sigma = 1, ncp = NULL)

Arguments

df	degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.
mu	The location parameter of the distribution. If ncp == 0 (or NULL), this is the median.
sigma	The scale parameter of the distribution.
ncp	non-centrality parameter \delta; currently except for rt(), accurate only for abs(ncp) <= 37.62. If omitted, use the central t distribution.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_student_t.html

In the following, let X be a location-scale Student's T random variable with df = \nu, mu = \mu, sigma = \sigma, and ncp = \delta (non-centrality parameter).

If Z follows a standard Student's T distribution (with df = \nu and ncp = \delta), then X = \mu + \sigma Z.

Support: R, the set of all real numbers

Mean:

For the central distribution (ncp = 0 or NULL):

E(X) = \mu

for \nu > 1, and undefined otherwise.

For the non-central distribution (ncp \neq 0):

E(X) = \mu + \delta \sqrt{\frac{\nu}{2}} \frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)} \sigma

for \nu > 1, and undefined otherwise.

Variance:

For the central distribution (ncp = 0 or NULL):

\mathrm{Var}(X) = \frac{\nu}{\nu - 2} \sigma^2

for \nu > 2. Undefined if \nu \le 1, infinite when 1 < \nu \le 2.

For the non-central distribution (ncp \neq 0):

\mathrm{Var}(X) = \left[\frac{\nu(1+\delta^2)}{\nu-2} - \left(\delta \sqrt{\frac{\nu}{2}} \frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)}\right)^2\right] \sigma^2

for \nu > 2. Undefined if \nu \le 1, infinite when 1 < \nu \le 2.

Probability density function (p.d.f):

For the central distribution (ncp = 0 or NULL), the standard t distribution with df = \nu has density:

f_Z(z) = \frac{\Gamma((\nu + 1)/2)}{\sqrt{\pi \nu} \Gamma(\nu/2)} \left(1 + \frac{z^2}{\nu} \right)^{- (\nu + 1)/2}

The location-scale version with mu = \mu and sigma = \sigma has density:

f(x) = \frac{1}{\sigma} f_Z\left(\frac{x - \mu}{\sigma}\right)

For the non-central distribution (ncp \neq 0), the density is computed numerically via stats::dt().

Cumulative distribution function (c.d.f):

For the central distribution (ncp = 0 or NULL), the cumulative distribution function is computed numerically via stats::pt(), which uses the relationship to the incomplete beta function:

F_\nu(t) = \frac{1}{2} I_x\left(\frac{\nu}{2}, \frac{1}{2}\right)

for t \le 0, where x = \nu/(\nu + t^2) and I_x(a,b) is the incomplete beta function (stats::pbeta()). For t \ge 0:

F_\nu(t) = 1 - \frac{1}{2} I_x\left(\frac{\nu}{2}, \frac{1}{2}\right)

The location-scale version is: F(x) = F_\nu((x - \mu)/\sigma).

For the non-central distribution (ncp \neq 0), the cumulative distribution function is computed numerically via stats::pt().

Moment generating function (m.g.f):

Does not exist in closed form. Moments are computed using the formulas for mean and variance above where available.

See Also

stats::TDist

Examples

dist <- dist_student_t(df = c(1,2,5), mu = c(0,1,2), sigma = c(1,2,3))

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

distributional documentation built on June 11, 2026, 9:07 a.m.