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

#> <distribution[3]>
#> [1] t(1, 0, 1) t(2, 1, 2) t(5, 2, 3)

#> [1] NA 1 2

#> [1] NA Inf 15

#> [[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
#>

#> [1] 0.06366198 0.14814815 0.12653556

#> [1] -2.754168 -1.909543 -2.067232

#> [1] 0.9220209 0.8638034 0.7327454

#> [1] 0.7265425 2.2344268 3.6782889