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

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):

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

Cumulative distribution function (c.d.f):

Nasty, omitted.

Moment generating function (m.g.f):

Undefined.

See also

Examples

set.seed(27) X <- StudentsT(3) X
#> Student's T distribution (df = 3)
random(X, 10)
#> [1] 1.4854556 -0.3809239 -1.8376741 0.1105147 0.3005249 0.1558420 #> [7] -1.5135073 -0.6088114 -2.4080689 -1.1878884
pdf(X, 2)
#> [1] 0.06750966
log_pdf(X, 2)
#> [1] -2.695485
cdf(X, 4)
#> [1] 0.9859958
quantile(X, 0.7)
#> [1] 0.5843897
### 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
#> [1] 1.378916
# 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))
#> [1] 0.2012211
# exactly equivalent to the above 2 * cdf(T, -abs(t_stat))
#> [1] 0.2012211
# p-value for one-sided test # H_0: mu <= 3 vs H_A: mu > 3 1 - cdf(T, t_stat)
#> [1] 0.1006105
# p-value for one-sided test # H_0: mu >= 3 vs H_A: mu < 3 cdf(T, t_stat)
#> [1] 0.8993895
### example: calculating a 88 percent T CI for a mean # lower-bound mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
#> [1] 2.631598
# upper-bound mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
#> [1] 6.368402
# equivalent to mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
#> [1] 2.631598 6.368402
# also equivalent to mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
#> [1] 2.631598
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
#> [1] 6.368402