StudentsT: Create a Student's T distribution In distributions3: Probability Distributions as S3 Objects

Description

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

Usage

 1 StudentsT(df)

Arguments

 df Degrees of freedom. Can be any positive number. Often called ν in textbooks.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

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

Support: R, the set of all real numbers

Mean: Undefined unless ν ≥ 2, in which case the mean is zero.

Variance:

ν / (ν - 2)

Undefined if ν < 1, infinite when 1 < ν ≤ 2.

Probability density function (p.d.f):

f(x) = Γ((ν + 1) / 2) / (√(ν π) Γ(ν / 2)) (1 + x^2 / ν)^(- (ν + 1) / 2)

Cumulative distribution function (c.d.f):

Nasty, omitted.

Moment generating function (m.g.f):

Undefined.

Value

A StudentsT object.