Stable lifecycle

dist_student_t(df, mu = 0, sigma = 1, ncp = NULL)

Arguments

df

degrees of freedom (\(> 0\), maybe non-integer). df = Inf is allowed.

mu

The location parameter of the distribution. If ncp == 0 (or NULL), this is the median.

sigma

The scale parameter of the distribution.

ncp

non-centrality parameter \(\delta\); currently except for rt(), only for abs(ncp) <= 37.62. If omitted, use the central t distribution.

Details

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.

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let \(X\) be a central 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):

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

See also

Examples

dist <- dist_student_t(df = c(1,2,5), mu = c(0,1,2), sigma = c(1,2,3)) dist
#> <distribution[3]> #> [1] t(1, 0, 1) t(2, 1, 2) t(5, 2, 3)
mean(dist)
#> [1] NA 1 2
variance(dist)
#> [1] NA Inf 15
generate(dist, 10)
#> [[1]] #> [1] 0.63017890 0.80817754 13.67350714 0.40719714 0.81023079 -0.07963269 #> [7] -0.61222199 -0.58856860 0.36477731 -1.47654233 #> #> [[2]] #> [1] -1.638119 3.561696 1.647123 -1.432021 -4.012675 1.650036 -1.397216 #> [8] 2.015674 6.345520 2.381649 #> #> [[3]] #> [1] 1.144954 5.356423 -1.954513 -6.163206 -2.044159 -7.777897 2.338265 #> [8] 6.399335 1.788889 4.995309 #>
density(dist, 2)
#> [1] 0.06366198 0.14814815 0.12653556
density(dist, 2, log = TRUE)
#> [1] -2.754168 -1.909543 -2.067232
cdf(dist, 4)
#> [1] 0.9220209 0.8638034 0.7327454
quantile(dist, 0.7)
#> [1] 0.7265425 2.2344268 3.6782889