distributions3 (version 0.1.2)

StudentsT: Create a Student's T distribution

Description

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.

Usage

StudentsT(df)

Arguments

df

Degrees of freedom. Can be any positive number. Often called $\nu$ in textbooks.

Value

A StudentsT object.

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 = $\nu$.

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):

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

Cumulative distribution function (c.d.f):

Nasty, omitted.

Moment generating function (m.g.f):

Undefined.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), Tukey(), Uniform(), Weibull()

Examples

# NOT RUN {
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)
# }