StudentsT | R Documentation |
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.
StudentsT(df)
df |
Degrees of freedom. Can be any positive number. Often called ν in textbooks. |
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.
A StudentsT
object.
Other continuous distributions:
Beta()
,
Cauchy()
,
ChiSquare()
,
Erlang()
,
Exponential()
,
FisherF()
,
Frechet()
,
GEV()
,
GP()
,
Gamma()
,
Gumbel()
,
LogNormal()
,
Logistic()
,
Normal()
,
RevWeibull()
,
Tukey()
,
Uniform()
,
Weibull()
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)
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