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

 StudentsT R Documentation

## Create a Student's T distribution

### 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

```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()`

### Examples

```
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)
```

distributions3 documentation built on Sept. 7, 2022, 5:07 p.m.