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.
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.
Examples
set.seed(27)
X <- StudentsT(3)
X
#> [1] "StudentsT 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