distributional (version 0.2.1)

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

Description

stable

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

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

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):

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

See Also

stats::TDist

Examples

# NOT RUN {
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)

# }