# 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.

Other continuous distributions: `Beta()`, `Cauchy()`, `ChiSquare()`, `Erlang()`, `Exponential()`, `FisherF()`, `Frechet()`, `GEV()`, `GP()`, `Gamma()`, `Gumbel()`, `LogNormal()`, `Logistic()`, `Normal()`, `RevWeibull()`, `Tukey()`, `Uniform()`, `Weibull()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56``` ```set.seed(27) X <- StudentsT(3) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7) ### example: calculating p-values for two-sided T-test # here the null hypothesis is H_0: mu = 3 # data to test x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2) nx <- length(x) # calculate the T-statistic t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx)) t_stat # null distribution of statistic depends on sample size! T <- StudentsT(df = nx - 1) # calculate the two-sided p-value 1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat)) # exactly equivalent to the above 2 * cdf(T, -abs(t_stat)) # p-value for one-sided test # H_0: mu <= 3 vs H_A: mu > 3 1 - cdf(T, t_stat) # p-value for one-sided test # H_0: mu >= 3 vs H_A: mu < 3 cdf(T, t_stat) ### example: calculating a 88 percent T CI for a mean # lower-bound mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) # upper-bound mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) # equivalent to mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) # also equivalent to mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx) mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx) ```