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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.

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):

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

Cumulative distribution function (c.d.f):

Nasty, omitted.

Moment generating function (m.g.f):

Undefined.

See also

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
#> [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