dist_student_t {distributional}

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(), only for abs(ncp) <= 37.62. If omitted, use the central t distribution.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let X be a central Students T random variable with df = \nu.

Support: R, the set of all real numbers

Mean: Undefined unless \nu \ge 2, in which case the mean is zero.

Variance:

\frac{\nu}{\nu - 2}

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

Probability density function (p.d.f):

f(x) = \frac{\Gamma(\frac{\nu + 1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} (1 + \frac{x^2}{\nu} )^{- \frac{\nu + 1}{2}}

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)